1
vote
1answer
30 views

Find a basis of $M_2(F)$ so that every member of the basis is idempotent

Let $V=M_{2\times 2}(F)$ (the space of 2x2 matrices with coefficients in a field $F$). Find a basis $\{A_1,A_2,A_3,A_4\}$ of $V$ so that $A_j^2=A_j$ for all $j$. My attempt. Let $A_j$ be ...
1
vote
2answers
30 views

Rank of $I_m - X_{m \times m}$ given rank of $X$

I have a matrix $X_{m\times m}$ which is idempotent and has $rank(X) = n < m$. I have for some time now been trying to calculate $rank(I_m - X)$ but have been unable to do so. I should be able to ...
2
votes
2answers
81 views

Are matrices vectors?

This may sound like an obvious question but it has confused me! According to wikipedia (https://en.m.wikipedia.org/wiki/Vector_(mathematics_and_physics)) vectors are defined as: "An element of a ...
4
votes
1answer
199 views

Null space basis

Let $V\in\mathbb{R}^{a\times b}$ be a matrix such that it is not a full column rank. Then there will be a nonsingular matrix $H$ such that $$VH=\left[\begin{array}{cc} V_{1} & 0_{a\times ...
0
votes
1answer
23 views

Rotating a plane defined by a normal and a distance from the origin around an arbitrary point in 3D space

I have a plane defined by its normal and its distance from the origin. I have a rotation matrix and a point in 3D space around which to do the rotation. What formula will allow me to do the rotation? ...
4
votes
1answer
77 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
0
votes
1answer
20 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
0
votes
1answer
37 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
0
votes
1answer
33 views

When does a matrix fail to be positive definite?

I am wondering how to think about a matrix being "bigger" than another. If I have the inequality X - Omega Sigma^-1 > 0 where all matrices are quadratic and X = Z'Z with Z positive definite and Omega ...
0
votes
1answer
34 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
1
vote
0answers
49 views

What do double vertical lines mean?

I am reading a paper on computer graphic and having hard time to understand this formula: What is the double vertical lines means? Do they always go with power of 2? If I want to learn further ...
0
votes
2answers
34 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
0
votes
1answer
47 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
1
vote
1answer
66 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
0
votes
0answers
17 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
0
votes
1answer
41 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
0
votes
3answers
44 views

Show that a linear matrix transformation is bijective iff A is invertible.

Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$. Show that it is bijective IFF $A$ is invertible. I was thinking then that I could show ...
0
votes
1answer
29 views

True/false with justification. $A ∈ M_{n}(K)$, and for any $v ∈ M_{n×1}(K)$, $ Av = 0$, so $A = 0$.

Let $A ∈ M_n(K)$. If for all $v ∈ M_{n×1}(K)$, $Av = 0$, then $A = 0$. True or false? Mark scheme states Yes because for all $ 1 ≤ i ≤ n$, we take $$v = ...
0
votes
1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
0
votes
1answer
22 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
1
vote
2answers
46 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
2
votes
3answers
324 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
2
votes
1answer
30 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
2
votes
2answers
47 views

closeness of a set of vectors

Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$ A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ...
0
votes
2answers
69 views

How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
0
votes
0answers
27 views

Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
0
votes
1answer
33 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
0
votes
1answer
37 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
0
votes
2answers
71 views

Is the determinant of a matrix some kind of “integral” of the linear mapping?

A $n \times n$ matrix corresponds to a linear mapping between two $n$-dim vector spaces. The determinant of a matrix gives a scalar, just as the integral of an integrable function gives a scalar. ...
1
vote
2answers
56 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
2
votes
2answers
53 views

How to bring $5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$ to the form $\langle x',Ax' \rangle = 1$

How can I bring $$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$ to the form $$\langle x',Ax' \rangle = 1$$ where $x' = \alpha x + \beta$ where $\alpha \in \mathbb{R}^+$ and $\beta \in ...
4
votes
1answer
33 views

$rk(A)=n$ implies $rk(AB)=rk(B)$

Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$. Assume $rk(A)=n$. Prove that $rk(AB)=rk(B)$. Lets start by proving $rk(B) \ge rk(AB)$. Indeed, since the ...
0
votes
1answer
40 views

$T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $ <A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $ <A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
1
vote
1answer
19 views

Vectors, columns and representations

When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column ($1\times M$ matrix) is not a vector, it's ...
0
votes
0answers
28 views

Matix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
0
votes
0answers
35 views

Write down a matrix of which only the null space is known?

What is the matrix in which null space are all of the multiples of the vector: $$\vec{v}=\begin{bmatrix}4 \\ 3 \\ 2 \\ 1\end{bmatrix}$$ I suppose there are a lot of solutions, but I don't I am not ...
0
votes
1answer
36 views

Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
0
votes
2answers
30 views

$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $

Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus H_2$ I stuck ...
1
vote
2answers
89 views

maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
1
vote
1answer
39 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
2
votes
1answer
48 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
0
votes
1answer
51 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
2
votes
1answer
39 views

linearly independent vectors and rows/cols space

Given $n$ vectors, we want to determine if those vectors are linearly independent. One way doing it is writing those vectors as columns of a matrix and row-reduce it. The vectors are linearly ...
3
votes
2answers
146 views

Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
1
vote
2answers
42 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
2
votes
0answers
59 views

kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...
2
votes
2answers
79 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
0
votes
1answer
15 views

Understanding the basis term

Consider: $$\left( {\matrix{ 0 & 1 & 2 \cr 0 & 0 & 0 \cr } } \right)$$ I want to find a basis for the row-space of the matrix above. One might say $$B = \left\{ {\left( ...
0
votes
0answers
88 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
1
vote
1answer
50 views

How do I rewrite vectors in other basis' given change of coordinate matrices?

$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,$\displaystyle ...