Let $V$ be the vector space over $\mathbb{R}$ whose basis is $(\sin(t),\cos(t))$. Let the scalar product be defined by $$\langle f,g \rangle =\int_{-\pi}^{\pi} f(t)g(t)\,\mathrm dt$$ What is the matrix of this scalar product with respect to the given basis?   $$\langle f,g \rangle = \int_\limits{-\pi}^{\pi} f(t)g(t)\,\mathrm dt = \int_\limits{-\pi}^{\pi}(\sin (t)+\cos(t))(\sin (t)+\cos(t))\,\mathrm dt = 2\pi$$  From a previous exercise with polynomials, I acquired this method. However I do not get a matrix but a number. Am I doing anything wrong? I do not know why people consider it a duplicate, if I am not getting the same result using the same method.   Thanks in advance.
124076
Let $V$ be the vector space over $\mathbb{R}$ consisting of all polynomials of degree $\leqslant n$. If $f,g\in V$, let $$\langle f,g \rangle = \int_\limits{0}^{1}f(t)g(t)dt$$ Find the matrix of this scalar product with respect to the basis $\{1,t,\dots,t^n\}$.  What does it mean to find a matrix of the scalar product? Thanks in advance!
63028
Suppose we have vector spaces $\mathbb W$ and $\mathbb V$.  A given basis for $\mathbb W$ is $\{(1, 1, 0, -1), (0, 1, 3, 1)\}$, and a given basis for $\mathbb V$ is $\{(1, 2, 2, -2), (0, 1, 2, -1)\}$.  How to find a basis for $\mathbb W\cap\mathbb V$?
69862
I'm trying to understand the UMP of the product functor as right adjoint to the diagonal functor but I'm having difficulties understanding the notation in Steve Awodey's . On page 210 he uses $(f_1, f_2)$, $\langle f_1, f_2 \rangle$ and $(f_1 \times f_2)$ all within one paragraph. What are those? How are these defined and how do they differ?
2257
Suppose that we are dealing with the real field $\mathbb{R}$ and we define $T:\mathbb R^2\to\mathbb R^2$ by $$T(u,v):=(-v,u).$$ It is trivial to see that $\langle(u,v),T(u,v)\rangle = 0$ for all $u, v \in \mathbb{R}$, where $\langle \, \cdot \, , \, \cdot \, \rangle$ is the standard inner product.  A question in Hoffmann-Kunze asks us to define all possible inner products $[ \, \cdot \, , \, \cdot \, ]$ such that $[\alpha, T(\alpha)]=0$ for all $\alpha\in\mathbb R^2$ and $T$ is as above. How does one find all such inner products?
7464
Given a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as $G_{i,j}=v_i \cdot v_j$  Due to symmetry in the dot product, $G$ is Hermitian.  I'm trying to remember why $|G|=0$ iff the set of vectors are not linearly independent.
13488
I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link:    It seems that with a matrix multiplication $AB=C$, that the entries as scalars, are formed from the dot product computations of the rows of $A$ with the columns of $B$. Visual interpretations from mechanics of overlpaing forces come to mind immediately because that is the source for the dot product (inner product).  I see the rows of $C$ as being the dot product of the rows of $B$, with the dot product of a particular row of $A$. Similar to the above and it is easy to see this from the individual entries in the matrix $C$ as to which elements change to give which dot products.  For understanding matrix multiplication there is the geometrical interpretation, that the matrix multiplication is a change in the reference system since matrix $B$ can be seen as a transormation operator for rotation, scalling, reflection and skew. It is easy to see this by constructing example $B$ matrices with these effects on $A$. This decomposition is a strong argument and is strongly convincing of its generality. This interpreation is strong but not smooth because I would find smoother an explanation which would be an interpretation begining from the dot product of vectors and using this to explain the process and the interpretation of the results (one which is a bit easier to see without many examples of the putting numbers in and seeing what comes out which students go through).  I can hope that sticking to dot products throughout the explanation and THEN seeing how these can be seen to produce scalings, rotations, and skewings would be better. But, after some simple graphical examples I saw this doesn't work as the order of the columns in matrix $B$ are important and don't show in the graphical representation.   The best explanation I can find is at . It is convincing but a bit disappointing (explains why this approach preserves the "composition of linear transformations"; thanks @Arturo Magidin). So the question is: Why does matrix multiplication happen as it does, and are there good practical examples to support it? Preferably not via rotations/scalings/skews (thanks @lhf).
100325
In Dummit and Foote problem 12 of section 10.4, I need to show that if $V$ is some vector space over a field $F$ and $v,v'$ are nonzero elements of $V$, supposing we have that $v \otimes v' = v' \otimes v$ then this implies that $v = av'$ for some $a \in F$. I was trying to consider some basis $\mathcal{B} = \{e_i\}_{i\in I}$ for the vector space $V$, show that we can write $v$ and $v'$ as linear combinations of this basis elements and then argue that since we have $v \otimes v' = \sum_{i \in I}r_i e_i \otimes \sum_{i\in I} s_i e_i = \sum_{i\in I} s_i e_i \otimes \sum_{i \in I}r_i e_i = v' \otimes v$ then if we look at a particular element of the sum on LHS and RHS, namely $r_ie_i \otimes s_je_j = s_je_j\otimes r_i e_i$ then we can send the $r_i$ to the other side or do some manipulation of that sort and obtain the desired result.   I know this argument is not rigorous at all, so I am having a hard time doing it more rigorous (if this is a good approach) or actually attack this problem efficiently. I was wondering if this would be the right approach, like working with basis elements and so on or if there is some better way to look at this problem. Perhaps some use of universal property for modules would be helpful? Any help or suggestion with this problem is highly appreciated! Thanks so much!
94487
If $V$ is a vector space (not the zero vector space) over $\mathbb{R}$, and if $F$ is a finite field.  how could I show that it is not possible to define a new scalar multiplication of $F$ on $V$, in a way that $V$ with this scalar multiplication and the usual addition becomes a vector space over $F$?  Thanks in advance.
98145
It is given that $T$ is a linear transformation from $\mathbb R^n$ to $\mathbb R^n$ such that $T^2 = k T $  for some $k\in \mathbb R$.   Then, one or more of the options are true    $\|T(x)\| = |k| \|x\|$ for all $x\in \mathbb R^n$.  If $\|T(x)\| = \|x\|$ , for some non- zero vector $x\in \mathbb R^n$ then $ k =\pm 1 $.  $\|T(x)\|  &gt;  \|x\|$ for some non- zero vector $x\in \mathbb R^n$, then T is singular.  $ T = k I $ where I is an identity transformation.    Please suggest how to proceed.
36068
I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ element. The book on Quantum Mechanics by Shankar says that the inner product between a basis vector and itself is not one, why not? Why can't these basis vectors be normalized to one, only to the ?
40249
For a lie algebra $\mathbb{g}  $ we can define the adjoint representation as:  $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $  I am wondering how to we compute adjoint matrices with respect to a given basis?     For example, let the lie algebra be $ \mathbb{g}=sl_2(\mathbb{C}) $ with the basis as follows: $X=\begin{bmatrix} 0&amp;1\\0&amp; 0 \end{bmatrix}$,$Y=\begin{bmatrix} 0&amp;0\\1&amp; 0 \end{bmatrix}$ and $H=\begin{bmatrix} 1&amp;0\\0&amp; -1 \end{bmatrix}$  Let $ ad: \mathbb{g} \rightarrow End(\mathbb{g})  $ be the adjoint representation  How would I compute the matrices $ ad_X $, $ ad_Y $ and $ ad_Z$ relative to the above matrices?
101631
In the beginning of linear algebra courses, there are vectors in $\mathbb R^n$ and the dot product is introduced. We learn that if the dot product of two vectors is zero, then these vectors are called orthogonal and there is a right angle between them. Therefore, we understand perpendicularity from the definition of orthogonality.   In the last chapters, all previous notions are generalized as vector spaces and inner products are introduced. Then we learn vectors $\vec u$ and $\vec v$ are orthogonal if $\langle \vec u, \vec v \rangle=0$ in inner product spaces. Our previous knowledge guides us to seek a right angle between them.   Let $P$ be the vector space of first degree polynomials. Let $p(x)=ax+b$ and $q(x)=cx+d$. Define $\langle p(x), q(x) \rangle$ as  $$\langle p(x), q(x) \rangle=ac+bd$$  Then if we consider $p(x)=x-2$ and $q(x)=4x+2$, we find that $\langle p(x), q(x) \rangle=0$. But if we draw them, we see that there is $30.96$ degrees between them. So orthogonality in inner product spaces does not necessarily mean perpendicularity. If not perpendicularity, what should we understand from orthogonality in inner product spaces?
47113
$\langle A,B \rangle := Trace (AB^t)$ is an inner product over the vector space $M_n(\mathbb R)$ of all real matrices of size $n$ , I would like to know whether this inner-product has any Geometrical or Physical significance (interpretation) or not ? Please shed some light . Thanks in advance
56031
Where $ f: V \rightarrow V  $ is a linear operator on an inner product space $V$, I'm trying to show that $$ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $$  i.e. the Kernel of the adjoint is equal to the orthogonal complement of the image of f.  I taking the inner product of $f(v)$, where $v \in V$  and an arbitary element of the kernel, $k \in \mathrm{ker}(f^*) $  $$\langle f(v), k\rangle = \langle v, f^*(k)\rangle = \langle v, 0\rangle = 0$$  And then taking the inner product of $f(v)$ with $w$, where $w \in  \mathrm{im}(f)^{\perp}$  By definition  of the orthogonal complement to the image of f,   $$ \langle f(v), w\rangle = 0 $$  My question is, now that we have   $$ \langle f(v), k\rangle = \langle f(v), w\rangle = 0 $$  What can we conclude? I would like to conclude that because $w$ and $k$ are arbitary elements of their respective subspaces, that these must be equal, i.e. $ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $  However, I am not sure this is mathematically sound, is that conclusion valid or are there any technical barriers or additional statements that would have to be made to make that watertight?
36846
Can someone help me out on how to find the the matrix representation and show proof that it is a linear transformation?    It is the rodrigues roation formula  and the matrix representation should look like    where c and s are cos and sin
1217
Seems like I , I think I am missing something important.     Let $V$ be an $n$ dimensional inner product space ($n \geq 1$), and $T\colon\mathbf{V}\to\mathbf{V}$ be a linear transformation such that:         $T^2 = T$   $||T(a)|| \leq ||a||$ for every vector $a$ in $\mathbf{V}$;         Prove that a subspace $U \subseteq V$ exist, such that $T$ is the orthogonal projection on $U$.   Now, I know these things:   The fact hat $T^2 = T$ guarantees that $T$ is indeed a projection, so I need to prove that T is an orthogonal projection (I guess this is where $||T(a)|| \leq ||a||$ kicks in). To do this I can prove that:   For every $v$ in $ImT^{\perp}$, $T(v) = 0$ Alternatively, I can prove that for every $v$ in $ImT$ and $u$ in $KerT$, $(v,u)=0$. $T$ is  (according to Wikipedia) The matrix $A = [T]_{E}$ when $E$ is an orthonormal basis, is hermitian (this is equivalent to the previous point). What else?    I've been thinking about it for quite some time now, and I'm pretty sure there is something big I'm missing, again. I just don't know how to use the data to prove any of these things.  Thanks!
68556
Let $V$ be a complex vector space of dimension $n$ with a scalar product, and let $u$ be an unitary vector in $V$. Let $H_u: V \to V$ be defined as  $$H_u(v) = v - 2 \langle v,u \rangle u$$  for all $v \in V$. I need to find the characteristic polynomial of this linear operator, but the only way to find it that I know of is using the associated matrix of the operator.  I don't know how to find this matrix because I don't know how to deal with the scalar product. Is there some other way to find the characteristic polynomial? If not, how can I find the associated matrix of this linear operator?  Thanks in advance.