# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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; ...
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### 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 ...
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### 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 $W$ ...
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### 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--- ...
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### 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 ...
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Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ... 2answers 69 views ### How do you find a non zero vector in Linear Algebra? The question is; The vectors a_1 = (1, 1, 0) and a_2 = (1, 1, 1) span a plane in \Bbb R^3. Find the projection matrix P onto the plane, and find a nonzero vector b that is projected to zero. ... 0answers 27 views ### Finding the least square fit for 3 parameters in Linear Algebra I know how to find least square for y = mx+b when we have two parameters. But this question has 3 parameters, am trying to think of how to approach it but so far no success, I can't find any ... 1answer 33 views ### Finding the exponential relation between two 4x4 transition matrices Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given 4\times4 transition matrices, representing transitions in three dimensions with the fourth ... 1answer 37 views ### Show that \lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \} and \lambda_m = \max \{ Q(u) \mid \|u\| = 1 \} Let V a vector space over K and Q(u) = \langle u, Tu \rangle a quadratic form. T is a symmetric operator. The eigenvalues of T are sorted by size \lambda_1 < \dots < \lambda_m. How ... 2answers 71 views ### Is the determinant of a matrix some kind of “integral” of the linear mapping? A n \times n matrix corresponds to a linear mapping between two n-dim vector spaces. The determinant of a matrix gives a scalar, just as the integral of an integrable function gives a scalar. ... 2answers 56 views ### Write Matrix A to A = \sum_{i=1}^{3} \lambda_i P_i Let A be a Matrix:$$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$$I ... 2answers 53 views ### How to bring 5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31 to the form \langle x',Ax' \rangle = 1 How can I bring$$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$to the form$$\langle x',Ax' \rangle = 1$$where x' = \alpha x + \beta where \alpha \in \mathbb{R}^+ and \beta \in ... 1answer 33 views ### rk(A)=n implies rk(AB)=rk(B) Let A \in Mat_{m\times n}(\mathbb{R}) and B \in Mat_{n\times p}(\mathbb{R}). Assume rk(A)=n. Prove that rk(AB)=rk(B). Lets start by proving rk(B) \ge rk(AB). Indeed, since the ... 1answer 40 views ### T : M_{n \times n}(R) \rightarrow M_{n \times n}(R) and T(A)= A^t and  <A,B> = Tr(AB^t) Let V = M_{n \times n}(R) with the inner product  <A,B> = Tr(AB^t), and T the linear operator given by T : M_{n \times n}(R) \rightarrow M_{n \times n}(R) and T(A)= A^t . How can i ... 1answer 19 views ### Vectors, columns and representations When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column (1\times M matrix) is not a vector, it's ... 0answers 28 views ### Matix column-wise multiplication operator I'm trying to find the proper operator for a column wise multiplication. Consider v=[v_1, v_2, ..., v_n]^T and A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ... 0answers 35 views ### Write down a matrix of which only the null space is known? What is the matrix in which null space are all of the multiples of the vector:$$\vec{v}=\begin{bmatrix}4 \\ 3 \\ 2 \\ 1\end{bmatrix}$$I suppose there are a lot of solutions, but I don't I am not ... 1answer 36 views ### Few basic things unclear to me about inner product spaces and orthonormal basis Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that:$$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$... 2answers 30 views ### T: V \rightarrow V a linear transformation such that T^2 = I and H_1= \{v \in V | T(v) = v\}\  and H_2= \{v \in V|T(v) = -v\}\  Let V a vector space and T: V \rightarrow V a linear transformation such that T^2 = I and H_1= \{v \in V | T(v) = v\}\  and H_2= \{v \in V|T(v) = -v\}\  then V = H_1 \bigoplus H_2 I stuck ... 2answers 89 views ### maximum and minimum dimension of the space generated by \{v_1,v_2,v_3,v_4\} I'm confused about this problem. I have four vectors v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0) with a,b real numbers. Determine the maximum and minimum dimension of the ... 1answer 39 views ### Left shift operator L: l^2 \rightarrow l^2 on the sequence space l^2$$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 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 ... 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 ... 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 ... 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 ... 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: ... 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 ... 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 ... 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( ...
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 ...
$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,\$\displaystyle ...