Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
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8 views

Properties of Hermitian and Positive Definite matrix

Let $A \in \mathbb{C}^{nxn}$ be Hermitian and positive definite. I have to show that $|a_{jk}|^2 < a_{jj}a_{kk}$ $max_{i,j=1,\dots,n}|a_{ij}| = a_{kk}$ for some $k$ with $1\leq k\leq n$ For ...
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27 views

Trace of a certain matrix

Let $A$ be a $227 \times 227$ matrix having distinct eigenvalues , with entries from $\mathbb Z_{227}$ , then what is the trace of $A$ ?
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33 views

Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear

I was asked to prove this statement. Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear It seems to me that for $k=1$ and $u=0$ the statement is proved. Is this correct? Many proofs use this ...
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Is there something called the Reduced Column echleon form?

I recently asked a question where I couldn't find the rank of a matrix. The question is : Problem on Finding the rank from a Matrix which has a variable At the time I believed in the answer, ...
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21 views

Why null space and column space?

I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my ...
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17 views

How to prove or understand this linear algebra assertion?

Given a matrix $B \in \mathbb{R}^{n \times k} $, and $B$ has rank $ k $. Therefore there exists a nonsingular matrix $A=( A_{1},A_{2}) \in \mathbb{R}^{n \times n} $ such that $$ AB= \left[ ...
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34 views

Shortest distance from a point to vertices of a cube

A $d$ dimension cube has vertices $P_1,...,P_{2^d}$, where the coordinates of each vertex are either $0$ or $1$. To find which vertex of $P_1,...P_{2^d}$ is closest to a given point $P=(p_1, ...
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How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$

How to do the part (iv) . Please help. Here are my answers to the first parts: (i) α a root of given equation $\implies \alpha^4-5 \alpha^2 + 2 \alpha -1 = 0$ $\implies \alpha^{n+4} - 5 ...
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28 views

Isometry problem [on hold]

An isometry $M : R^n → R^n$ is a map that preserves distance, i.e. $||M(v) − M(w)|| = ||v − w||$ for all $v, w ∈ R^n$ . • Let M be an isometry with $M(0) = 0$. Let $e_i$ be the $i$th standard unit ...
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Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
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29 views

Proving $A_{n}$ is not invertible for n>2 when the entries are sequential integers

Let $A_{n}$ be the nxn matrix whose entries are the integers 1, 2, 3,..., n-1, n, written in order from left to right, top to bottom. For example, $$A_{5}=\begin{bmatrix} ...
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1answer
25 views

$\operatorname{tr}(AB) = 0$ for (skew-)symmetric matricies

I know if A is symmetric and B is skew-symmetric then $\operatorname{tr}(AB) = 0$. (This follows because $\operatorname{tr}(AB) = -\operatorname{tr}(AB) $) Is the converse of that true? In other ...
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32 views

If A^2 =0 then possible rank of A

Let, A be a non zero matrix of order 8 with A^2 =0. Then one of the possible value for rank of A is (a) 5 (b) 4 (c) 6 (d) 8
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Trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ takes on different shapes.

Note: This is a homework problem. I'm trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ is an ellipse or the empty set if $ac-b^2\gt 0$. There are others to show but if I can understand this ...
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8 views

Partial Sum to be invertible [on hold]

Let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$m>n, A_1+\cdots+A_m=E_n.$$ Show that there exists a subset $P\subset \{1,\cdots,n\}$ such that $$\sum_{k\in S}A_k$$ is invertible.
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Why is $\det(A-\lambda I)=(\lambda-c)^n$ when $(A-cI)^n=0$?

Let $A$ be a $n\times n$ matrix and suppose that $(A-cI)^n=0$ for some scalar $c$. Then why the characteristic polynomial of $A$ is $(x-c)^n$?
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25 views

Compute $B=QAQ^{-1}$

$A,B$ are $n\times n$ matrices, $B=QAQ^{-1}$, and I know $A$ and $B$, how to compute $Q$? I know if $T$ a linear transformation, and with different basis we get $A$ and $B$, and we could use these ...
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6 views

Number of pivot columns in a 4x6 matrix for spanning set to occur

How many pivot columns must a 4x6 matrix have if its columns span $\mathbf{R}^4$? Explain. So, in my head, this is pretty clear: You need four dimensions => So you need a minimum of four vectors that ...
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1answer
39 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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Linear Algebra Analytical Exercise

This one has me stumped... $$H=C(sI-A)^{-1}B$$ and $$H_{CL} = C(sI-A+BK)^{-1}BG$$ Show that $$H_{CL} = H[I+K(sI-A))^{-1}B]^{-1}G$$ Any hints would be greatly appreciated!
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Find a basis for symmetric $2 \times 2$ matrices [on hold]

Find a basis for the space of all $2 \times 2$ symmetric matrices. I do not even know how to start. please explain it to me step by step
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1answer
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Solving symbolic linear equations with maple

How can I solve linear equations of the following type in Maple? $$\begin{pmatrix} 1 & 1 & 1 & 1\\ b-c & c-b & a-b &0 \\ b-d & d-a & 0 &a-b \end{pmatrix} ...
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8 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
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1answer
22 views

Simple Eigenvalue finding question (by gauss elimination)

I saw a method for finding eigenvalues by using Gauss elimination to find an upper triangular matrix, then just taking the diagonal elements as the eigenvalues. It seems to work except for this case: ...
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13 views

Subspace vector proofs problem [on hold]

I'm having trouble understanding/solving this proof. QUESTION: Prove the set P_3 is a subspace of P_4 with standard operations, where P_n is a vector space of all polynomial functions with degree n ...
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1answer
21 views

Basis for vector space $\mathbb{R}^{m\times n}$

My question is whether my solution to the following problem is valid. The problem is from Artin's Algebra, chapter 3: Let $(X_1,\cdots,X_m)$ and $(Y_1,\cdots,Y_n)$ be bases for $\mathbb{R}^m$ and ...
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22 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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32 views

What does determinant of linear operator mean?

I am solving problem (Linear Algebra by Hoffman, Excercise 5.4.8) : Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $B$ be a fixed element of $V$ and let $T_B$ be the ...
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Representation of Matrix with Rank 1

Prove that every $m \times n$ matrix of rank $1$ has the form $A=XY^t$, where $X,Y$ are $m$- and $n$-dimensional column vectors. How uniquely determined are these vectors$?$ My attempt: I thought ...
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Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...
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30 views

True or False Question About Linear Algebra

I 'm new in Stack and I need help with a few questions about linear algebra. I'm trying it but I cannot. TRUE OR FALSE 1) Let $A,B$ and $C$ be $nxn$ matrices such that $C$ is invertible and $B=C.A.{ ...
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how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
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1answer
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Linearization of a function in the point 0, 0

The linearization of the function $ f(x, y) = 1 + 2(x + 1) + 3(y + 1) + 4x^2 + 5y^2 $ in the point (0, 0) is given by: $ L(x, y) = 6 + 2x + 3y $ I know this is true, but how does one come to this ...
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1answer
17 views

Range and Nullspace of a transformation

Let $$(Tf)(x)=\int_0^xf(t)dt$$ be a transformation from the vector space V of all functions from $\mathbb{R}$ to $\mathbb{R}$ which are continuous. Describe the range and nullspace $T$. To me it ...
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Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
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22 views

There are at least four 3-dimensional subspace in $R^4$?

Shouldn't there be EXACTLY four 3-dimensional subspaces in $R^4$? My reasoning is that 3-d subspaces occur in $R^4$ when the rank of the augmented matrix of {$c_1:c_2:c_3:c_4$} is 3?
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23 views

What does $b^*$ mean?

What is this notation, my book explains nothing of it. I've colored it in yellow! I am guessing it stands for $b^{-1}$ or $b^1$?
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31 views

Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$? [duplicate]

Basically does a vector set that is linearly independent in $R^n$ automatically span $R^n$? My initial thought is yes, but is there some counterexample that can disprove this?
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Markov chains: identifying a nonregular transitional matrix

I am currently TA'ing for a course in which the students are soon to learn about Markov chains and stochastic matrices. During the sections, it refers to the possible existence of a stable state and ...
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Prove that the determinant of this matrix is non-zero.

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 ...
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1answer
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Some operation like determinant

we have determinant operation that is like below: $ det(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}) $= $ (-1)^{1+1}a(ei-fh)+ ...
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25 views

Every basis in the space of matrices 2x2 contains a non invertible matrix?

My initial thought that this was true; however, I thought of this counter example that may be possible? Would something like: ...
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Finding an upper bound for solution of $Ax=b$

Is there any upper bound on $x$ in $ Ax=b $ using some features of $A$, for example $ \min(\lambda_i) $ or $ \max(\lambda_i) $ eigenvalues of $A$ ? I've tried to find something like $$ \Vert x\Vert ...
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find the intersecting line in the provided planes $( 3x+2y+z = -1 ; 2x-y+4z=5)$

find the intersecting line in the provided planes $(3x+2y+z = -1 ; 2x-y+4z=5)$. I keep getting the wrong answer, here is my approach: I set it up in an augmented matrix: $$ ...
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1answer
9 views

Find an orthonormal basis for W and $W^{\perp}$

Consider $\mathbb{C}^3$ with the standard inner product (that is, the dot product), and let $W = \text{span} \{(1, 0, 1), (i, i, i)\}$. (a) Find an orthonormal basis for $W$. (b) Find an orthonormal ...
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1answer
13 views

Statement about non-homogeneous linear system with $n+1$ equations and $n$ unknowns

Let $M$ be some non-homogeneous linear system of $m$ equations and $n$ unknowns and $m=n+1$. Is it true that if the row echelon form of the augmented matrix (extended coefficient matrix) of $M$ ...
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30 views

Determine the values of c for which the equation Ax = b is consistent.

Determine the values of c for which the equation Ax = b is consistent. A= ...
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1answer
23 views

Concerning crossproduct and orthonormality of vectors

If you have 3 orthonormal vectors, $\pmatrix{u_1 \\ u_2 \\ u_3}$, $\pmatrix{v_1 \\ v_2 \\ v_3}$, $\pmatrix{w_1 \\ w_2 \\ w_3}$ such that $u= v \times w$ (crossproduct) can it be true that $v_i v_j + ...
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3answers
187 views

Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...