Tagged Questions
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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0answers
9 views
least squares minimization problem
It's easy to show that the solution to a least squares problem such as minimizing $||Ax+b||$ is $(A^tA)^{-1}A^tb$.
But how can one minimize $\sum_{i}||A_ix+b_i||$?
In one of the passages of the ...
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2answers
21 views
If $T$ is the derivative operator and $(v_0,…,v_m)$ is a standard basis, how can I prove that $Tv_k \in {\rm span}(v_0,…,v_k)$?
If I have that:
$p \in \mathbb{P}_m(\mathbb{C})$ is the set of all polynomials with complex coefficients with degree less than or equal to m.
And that $T$ is the differentiation operator:
$Tp=p'$
...
1
vote
2answers
23 views
Number of triangles in a Graph/Network
Given An undirected graph/Network, and its adjacency matrix A, and 1 (A column vector with all elements as 1).
How do we represent the problem of finding the number of triangles in the network ...
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3answers
25 views
Simple Trigonometry and algebra
If $$\sec\theta = X + \frac{1}{4X},$$
then what is $${\sec\theta + \tan\theta}$$ in terms of $X$?
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2answers
36 views
$A^m = r_m(A)?$ Power of a matrix!
In my Linear Algebra textbook we are reading, the following is stated for computing the power of a matrix in one of the advanced chapters as an exercise, $A^m = r_m(A)$. $r_m (A)$ is the remainder ...
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vote
5answers
76 views
Showing something is not onto?
Quick question..:
If I have a linear transformation $T:\mathbb R^2 \to \mathbb R^4$, can I say that since $\mathbb R^4$ is greater than $\mathbb R^2$, it can never be onto - since the elements in ...
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2answers
57 views
How to find $\det(-6A)$, if $\det A=-4$?
How do I solve this?
Assume that $A$ and $B$ are $6 \times 6$ matrices, such that $\det A = -4$ and $\det B = -2$. Find $\det(-6A)$.
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0answers
21 views
Relation between softmax and max
For two vectors $X$ and $Y$ in $\mathbf{R}^n$, does the inequality below hold?
$\left| \text{softmax} X - \text{softmax} Y \right|
\leq
\text{max} | X - Y |$
Softmax is the same as log-sum-exp:
...
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1answer
12 views
If T is in the set of complex polynomials, prove that the matrix of T w.r.t. the standard basis of T is upper triangular
Suppose that we have that $p \in \mathbb{P}_m(\mathbb{C})$ is the set of all polynomials with complex coefficients with degree less than or equal to m.
And that $T$ is the differentiation operator:
...
3
votes
2answers
56 views
Why is this a good picture of a covector?
I'm reading a book about applied differential geometry and the author says: "suppose $V$ is a finite dimensional vector space. For a given covector $\omega \in V^\ast$, the set $\hat{\omega}$, of ...
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3answers
62 views
Determine whether the set is a vector space.
So I have a final tomorrow and I have no clue how to determine whether a set is vector space or not. I've looked online on how to do these proofs but I still don't understand how to do them. Can any ...
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1answer
28 views
Let $V$ be a finite dimensional vector space over a field $F$ and $W_1,W_2$ be two subsapces then $(W_1+W_2)^0=W_1^0\cap W_2^0.$
Let $V$ be a finite dimensional vector space over a field $F$ and $W_1,W_2$ be two subsapces then
$(W_1+W_2)^0=W_1^0\cap W_2^0.$
8
votes
1answer
77 views
Prove a $n \times n $ matrix has rank 3
I have been examining a problem dealing with finding the rank of a $n \times n $ matrix $M$ as follows:
\begin{bmatrix}
0&1&4&9&16&\cdots &(n-1)^2\\
...
1
vote
1answer
18 views
Let $V$ be a finite dimensional vector space and $T$ be linear operator on $V$
Let $V$ be a finite dimensional vector space and $T$ be linear operator on $V$ such that $rank(T)=rank(T^2)$.Then to prove that the null space and range space of $T $are disjoint, i.e. zero vector is ...
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votes
3answers
44 views
Find eigenvalues of $T$ if $T(w,z)=(z,w)$
I need to find the eigenvalues of the above Transformation. Is it as easy as recognizing that the matrix of $T$, $M(T)=\begin{bmatrix}0&1\\1&0\\\end{bmatrix}$?
