Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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1answer
29 views

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 \alpha^...
3
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1answer
113 views

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, i....
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0answers
40 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} 1&2&3&4&5\\6&...
1
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1answer
73 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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2answers
143 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) $8$. Attempt : As , $A^2=0$ , so $A$ is a nilpotent ...
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0answers
26 views

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 ...
3
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1answer
127 views

Partial Sum to be invertible

Let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$m>n, A_1+\cdots+A_m=E_n,$$ where $E_n$ is the $n\times n$ identity matrix. Show that there exists a subset $P\subset \{1,\cdots,m\}$ ...
3
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2answers
109 views

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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3answers
831 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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2answers
782 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 ...
0
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1answer
33 views

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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1answer
588 views

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} \...
2
votes
1answer
1k 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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1answer
64 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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1answer
88 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 ...
2
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4answers
211 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 ...
1
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2answers
126 views

Representation of Matrix with Rank 1 [duplicate]

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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0answers
71 views

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 ...
0
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1answer
64 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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1answer
131 views

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: ...
0
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1answer
75 views

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
34 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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0answers
103 views

Is High Maturity and Proficiency in Calculus and Linear algebra necessary for successful research?

Is it the case that most successful mathematicians have very high Maturity and Proficiency(do it without thinking) in Calculus and Linear algebra, both calculation part(double/line integrals, ...
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1answer
33 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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1answer
34 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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3answers
579 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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1answer
51 views

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
20 views

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)+ (-1)^{1+2}b(di-fg)+(-1)^{...
0
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1answer
123 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: {$\begin{bmatrix}1&1\\0&1\end{bmatrix},\begin{bmatrix}1&0\\1&...
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0answers
46 views

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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2answers
22 views

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: $$ \left[\begin{array}{rrr|r}...
1
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1answer
419 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
45 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 where $m=n+1$. Is it true that if the row echelon form of the augmented matrix (extended coefficient matrix) of $M$ ...
0
votes
2answers
163 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= \begin{pmatrix}3&2&5&-1&-4\\2&1&4&2&-4\\1&-3&9&-2&15\\4&2&8&1&...
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1answer
35 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
617 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 $\mathrm{Tr}(A) = 0$. ...
0
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1answer
20 views

Find vectors $(a,b)$ and $(c,d)$ so that the frame $\{(1,1),(1,-1),(c,d),(a,b)\}$ is PRR equivalent to the frame

Find vectors $(a,b)$ and $(c,d)$ so that the frame $\{(1,1),(1,-1),(c,d),(a,b)\}$ is PRR equivalent to the frame $\{(\sqrt{2},0),(0,\sqrt{2}),(\sqrt{\frac{4}{3}},\sqrt{\frac{2}{3}}),(-\sqrt{\frac{2}{3}...
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2answers
52 views

Linear Algebra Subspace question

Here is the following question and answer for the question. I don't seem to quite grasp the answers or how the answers are what they are. Requirements for subspace: the zero vector is in the subset ...
3
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3answers
386 views

Relation between trace and rank for projection matrices

If $A $ is an $n \times n$ matrix over $\mathbb C$ such that $A^2=A$ then is it true that $\operatorname{trace} A = \operatorname{rank} A$?
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1answer
71 views

Is the pair controllable/observable?

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and $(Q,...
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5answers
69 views

Whether a matrix is a zero matrix

If a real square matrix $A $ is similar to a diagonal matrix and satisfies $A^n=0$ for some $n\in \mathbb N $,then can it be proved that $A$ must be a zero matrix?
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1answer
29 views

Bases and Matricies

Assume $ dim V = 2 $ and $\{e_1, e_2\}$ is a basis of $V$. Suppose that $$ M(T,\{e_1,e_2\}) = \left (\begin{array}{cc} 0 & 0 \\ 1 & 1 \end{array} \right) $$ Find a basis $\{v_1,v_2\}$ of $V$ ...
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0answers
44 views

Tensor product of two linear map and its matrix representation

Suppose $T_1: \mathbb{R}^n\to\mathbb{R}^n$ be any linear map and wrt a basis $\{e_1,\dots,e_n\}$ the matrix of $T_1$ is $M$, aand $T_2:\mathbb{R}^m\to\mathbb{R}^m$ be another linear map whose matrix ...
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1answer
66 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
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1answer
70 views

Problem on Finding the rank from a Matrix which has a variable

$$ A = \begin{bmatrix} 1 & -1 & -2 & -3 \\ -2 & 1 & 7 & 2 \\ -3 & 3 & 6 & \alpha \\ 7 & -6 & -17 & -17 \end{bmatrix} $$ Find the rank when $\alpha=...
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1answer
262 views

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$?

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$ ? I've been trying to sketch a proof by induction, but it seems more complicated that it should ...
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0answers
48 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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1answer
1k views

Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then ...
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1answer
37 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
4
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0answers
351 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...