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 ...

learn more… | top users | synonyms

-1
votes
1answer
26 views

Help please. System of equations.

Which method is much easier to determine the solution of these two system of equations? 1) y = 7/5x - 9 2) 4y = 3x + 3
0
votes
2answers
44 views

Matrix question help.

Consider $$X = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}.$$ Find a real matrix $A$ for which $A^2 = X$. I don't know how to answer this or where to start. ...
0
votes
1answer
89 views

Show set of vectors is a subspace of R^3

Problem: Determine which of the following are subspaces of ${\bf R}^3$. All vectors of the form $(a, b, c)$, where $b = a + c + 1$. My answers: Thought process: to show if a set of vectors called W ...
0
votes
1answer
47 views

LITERATURE (( Linear algebra [Intermediate level] ))

Good day people, Please, can somebody suggest me some literature on this specific topic: Linear algebra [Intermediate level] Thanks so much in advance.
2
votes
1answer
172 views

How to put this matrix in Jordan Canonical Form

Suppose you have the matrix $A = \begin{bmatrix} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$ Put it into ...
0
votes
1answer
92 views

Linear algebra [Intermediate level]

I was wondering how to solve a linear algebra problem. At first sight it looks easy. But 'between lines'. it isn't. The problem: Giving these matrices that are inversible, square (n x n) and are ...
2
votes
2answers
40 views

rank of a matrix with two columns s.t. their dot product is zero

I have function $\sigma(u,v)=(f(u,v),g(u,v),h(u,v))$ s.t. $\sigma_u$ x $\sigma_v\neq(0,0,0)$ (cross-product) also, there is the $3\times 2$ matrix : $$ ...
0
votes
2answers
17 views

Defining and expressing as a system of two equations. Is my answer good?

We wish to spend $\$164.00$ by purchasing $10$ books, some costing $\$15.00$ and other $\$17.00$. How many books of each price do we buy? My answer: let $x$ = number of books costing $\$15.00$ and ...
1
vote
1answer
28 views

Trouble understanding finite vector spaces and Gaussian coefficent

I have studied linear algebra for 2 months now and i cannot understand a task that i am currently trying to solve. Basically i am trying to find the amount of bases for n-dimensional vector space over ...
0
votes
1answer
43 views

Finding the R.E.F of an augmented matrix in a multiple choice exam

If I am given an augmented matrix and asked to fin it's reduced echelon form in a multiple choice paper, i.e, I am given some ref's and asked which one is the corresponding one for the augmented ...
0
votes
0answers
63 views

Isomorphism Linear Algebra

I'm currently going through a proof and I've come across something I don't really understand: Next, an endomorphism of a left $A$-module $M$, over a ring $A$ is an $A$- homomorphism ...
2
votes
0answers
29 views

Classroom enhancement resources for linear algebra

My university offers small grants (in the \$100 - \$300 range) for classroom enhancement resources. In the past, I've applied for and received a grant to purchase a Zometools kit used for ...
0
votes
3answers
50 views

Rotation of a vector

My linear algebra textbook only explained the rotation of a vector in a counterclockwise direction. I'm just wondering what happens if I rotate a vector in the clockwise direction? Do I solve such ...
0
votes
1answer
53 views

Calculating Cholesky decomposition directly via Cholesky of submatrix

I want to represent a Cholesky decomposition of $(n+1) \times (n+1)$ matrix $B$ which is of the form $B = \begin{bmatrix} k& v^T \\ v&A \end{bmatrix}$ where $k>0$, $v \in \mathbb{R}^{n}$ ...
0
votes
1answer
33 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
0
votes
2answers
86 views

Linear Equation system - solve only solvable variables

If I have the SVD of a Matrix $A$, how do I solve the linear equation system $Ax=b$? The problem is that if I e.g. has this linear equation system: $-2y + z = 3$ $-4y + 2z = 6$ $x -2y + z = 4$ ...
0
votes
1answer
35 views

What is the value of $a + b + c + d$ if the following equation holds?

If $a, b, c$ and $d$ are positive integers less than $7$ and $$a(7)^3 + b(7)^2 + c(7) + d = 901$$ What is the value of $a + b + c + d$? Is it related to consum of roots and product of roots?
3
votes
3answers
95 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
0
votes
1answer
13 views

Projection with modulus less than one

Let $X$ be an Hilbert Space, $X=Y\bigoplus Z$ where $Y$, $Z$ are both closed subspaces. Let $P:X \rightarrow X$ $P(y+z)= y$ be the canonical projection, then $||P|| \leq 1 \implies Y=Z^{\bot}$ ...
0
votes
1answer
82 views

Finding basis of vector spaces

Without proof find the dimension and a basis of the following vector spaces $V$ over the given field $K$. $V$ is the set of all polynomials over $\mathbb{R}$ of degree at most $n$, in which the sum of ...
1
vote
3answers
100 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
0
votes
3answers
60 views

For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...
0
votes
1answer
149 views

norm of linear operator less than or equal to abs of eigenvalue

$A: V\longrightarrow V$ linear operator V finite dimensional inner product space (1) Show that $|b|$ is less than or equal to $||A||$ where b is any eigenvalue of $L$. (2) Now let $L$ be ...
1
vote
1answer
102 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
0
votes
1answer
59 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
1
vote
1answer
48 views

Linearize a specific equation

Is it possible to linearize this equation? I tried without success. $$Y = \dfrac{a \cdot L1 \cdot \left(L1 \over L2\right)^X-L2}{a \cdot \left(L1 \over L2\right)^X-1}$$ I want something in $Y' = AX' ...
0
votes
2answers
37 views

How to calculate the determinant when the diagonal is in terms of $k$?

Determine the value of $k$ so that the columns in this matrix are linearly dependent: $$\begin{bmatrix} k & -1/2 & -1/2\\ -1/2 & k & -1/2\\ -1/2 & -1/2 & k ...
1
vote
1answer
29 views

finding a power of a matrix

When you are given the eigenvectors and eigenvalues of a matrix A and are asked to solve for A^3, for the formula A = PDP^-1 is your diagonal matrix the identity matrix with the eigenvalues swapped ...
0
votes
2answers
32 views

How to determine column dependency without calculating the determinant?

Determine whether this matrix' columns are linearly dependent or not. $$\begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}$$ The determinant is $0$ ...
1
vote
1answer
34 views

Orthogonal projection onto a plane

Find the minimal distance from the point $P = \begin{bmatrix}\\ -8 \\ 14 \\ 8 \end{bmatrix}$ to the plane $V$ of $\mathcal{R}^3$ spanned by $\begin{bmatrix}\\ 1 \\ 2 \\ -2 \end{bmatrix}$ and ...
0
votes
1answer
76 views

Sesquilinear forms seen as bilinear maps

Let $V$ be a complex vector space. A sesquilinear map (or conjugate-linear in the first variable and linear in the second) on a complex vector space $V$ is a map $f: V \times V \rightarrow \mathbb{C}$ ...
4
votes
4answers
110 views

Besides being symmetric, when will a matrix have ONLY real eigenvalues?

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation ...
3
votes
3answers
170 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
1
vote
1answer
29 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
0
votes
0answers
401 views

How to calculate a linear combination for a matrix' column?

I have a very weak understanding of linear dependency and linear combination, so I figured I'd check out some exercise about it: $$A = \begin{bmatrix} 4 & 0 & 1\\ 2 & 3 & 6\\ 6 ...
0
votes
1answer
99 views

Explain why each set is NOT a basis for the given vector space

My biggest problem with linear algebra is trying to get the wording right when I answer questions. I want to communicate my answers as effectively as possible. So here are my answers to the following ...
1
vote
0answers
36 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
1
vote
1answer
19 views

Going from $X^tAB - I = X^t$ to $X^t(AB-I) = I$ in matrix algebra.

Finding the value of the matrix $X$: $$X^tAB - I = X^t$$ I noticed that the next step chosen by my book is $$X^t(AB-I) = I$$ It's not clear to me how did they reach that. How did they go from one ...
2
votes
2answers
145 views

Algebraic and Geometric Multiplicity

I am having a hard time understand these two concepts Algebraic multiplicity and Geometric multiplicity of a matrix regarding its eigenvalues for example if I have the matrix: ...
1
vote
1answer
47 views

Proving vector projection

For nonzero vectors, how do you prove the following? $$\|u\|^2 = \|\text{projection of $u$ onto $v$}\|^2 + \|u - \text{projection of $u$ onto $v$}\|^2$$ I think what we need to do is split up the ...
1
vote
0answers
25 views

I need help understanding what r-th and s-th rows are.

Let E be the matrix obtained from the unit $n \times n$ matrix by multiplying the $r$-th row with a number $c$ and adding it to the $s$-th row, $r \neq s$. Let $A$ be an $n \neq n$ matrix. Then ...
0
votes
1answer
32 views

Calculating matrix determinants based on another's.

$$A = \begin {bmatrix} a & b & c \\ 4 & 0 & 2 \\ 1 & 1 & 1 \end {bmatrix} \ \ , \ \ \left| \ A \ \right| = 3$$ Knowing only this, how does someone calculate the determinant ...
0
votes
2answers
108 views

$3 \times 3$ matrix with entries $-1$ and $1$

There are $512$ matrix due to $2^9$. Is there a way instead of by hand to find how many of the matrix may equal $1, 2, 3....,$ etc. with the entries being $1$ and $-1$? Thanks in adavance.
0
votes
1answer
141 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
2
votes
2answers
160 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
1
vote
1answer
25 views

Can we list down all order 4 integer valued 3 x 3 matrices

Can we list down all integer 3 x 3 matrices($A$) whose are order 4 i.e $A^4= I$? or atleast get some examples? What should be the method for such thing?
1
vote
3answers
213 views

Creating and solving large systems of equations

I am trying to follow a solution in a book so that I can build my own model. They produce the set of equations below. The book claims it to be a system of equations with 10 unknowns; however from my ...
1
vote
1answer
57 views

null space of an n-by-m matrix

I have an $n$-by-$m$ ($n>m$) matrix named $J$. I wanted to find its null space so as I used matrix $M$ defined bellow: $$JM=0\text{, when } M=I-J^\dagger J$$ $J^\dagger$ is the pseudo inverse of ...
0
votes
0answers
45 views

Calculus and Matrices

Suppose I have a linear operator $T: \mathbb{R} \rightarrow \mathbb{R}$, and also suppose that it's a composition of elementary functions, so its derivative, $T'$, is reasonable easy to find. I can ...
0
votes
2answers
52 views

orthogonal complement problem: show $\operatorname{oc}(A\cap B)=\operatorname{oc}(A)+\operatorname{oc}(B)$

$A$ and $B$ are subspaces of $V$, a finite-dimensional inner product space. Show that $$\operatorname{oc}(A\cap B)=\operatorname{oc}(A)+\operatorname{oc}(B)$$