Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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How to get a reflection vector?

I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be? In the below image, I have d and n. How can I…
Nick Heiner
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100
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8 answers

Union of two vector subspaces not a subspace?

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
NSjonas
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66
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3 answers

Necessity/Advantage of LU Decomposition over Gaussian Elimination

I am reading the book "Introduction to Linear Algebra" by Gilbert Strang and couldn't help wondering the advantages of LU decomposition over Gaussian Elimination! For a system of linear equations in the form $Ax = b$, one of the methods to solve the…
62
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5 answers

Is there a definition of determinants that does not rely on how they are calculated?

In the few linear algebra texts I have read, the determinant is introduced in the following manner; “Here is a formula for what we call $\det A$. Here are some other formulas. And finally, here are some nice properties of the determinant.” For…
56
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5 answers

Best Fitting Plane given a Set of Points

Nothing more to explain. I just don't know how to find the best fitting plane given a set of $N$ points in a $3D$ space. I then have to write the corresponding algorithm. Thank you ;)
G4bri3l
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56
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4 answers

Why $\{\mathbf{0}\}$ has dimension zero?

According to C.H. Edwards' Advanced Calculus of Several Variables: The dimension of the subspace $V$ is defined to be the minimal number of vectors required to generate $V$ (pp. 4). Then why does $\{\mathbf{0}\}$ have dimension zero instead of one?…
Qingtian
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52
votes
4 answers

Finding the basis of a null space

I am trying to understand why the method used in my linear algebra textbook to find the basis of the null space works. The textbook is 'Elementary Linear Algebra' by Anton. According to the textbook, the basis of the null space for the following…
Sara
  • 1,249
52
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4 answers

Proving the trace of a transformation is independent of the basis chosen

How would you prove the trace of a transformation from V to V (where V is finite dimensional) is independent of the basis chosen?
Freeman
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46
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2 answers

A matrix is similar to its transpose

Possible Duplicate: Matrix is conjugate to its own transpose How can I prove that a matrix is similar to its transpose? My approach is: if $A$ is the matrix then $f$ is the associated application from $K^n\rightarrow K^n$. Define…
Mec
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41
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6 answers

why don't we define vector multiplication component-wise?

I was just wondering why we don't ever define multiplication of vectors as individual component multiplication. That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 \rangle$ to be $\langle a_1a_2, b_1b_2 \rangle$?…
dan
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40
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4 answers

How did mathematicians decide on the axioms of linear algebra

So we vector spaces, linear transformations, inner products etc all have their own axioms that they have to satisfy in order to be considered to be what they are. But how did we come to decide to include those axioms and not include others? For…
39
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2 answers

Proof of when is $A=X^TX$ invertible?

Say we have an $n\times m$ matrix $X$. What are the specific properties that $X$ must have so that $A=X^TX$ invertible? I know that when the rows and columns are independent, then matrix $A$ (which is square) would be invertible and would have a…
38
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11 answers

Linear independence of $\sin(x)$ and $\cos(x)$

In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if $0 = a\cos(x) + b\sin(x)$ implies that $a=b=0$,…
V. Galerkin
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36
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7 answers

Find the square root of a matrix

Let $A$ be the matrix $$A = \left(\begin{array}{cc} 41 & 12\\ 12 & 34 \end{array}\right).$$ I want to decompose it into the form of $B^2$. I tried diagonalization , but can not move one step further. Any thought on this? Thanks a lot! ONE STEP…
BVFanZ
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35
votes
8 answers

Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue.

I can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition: A matrix $A$ is singular if and only if $0$ is an eigenvalue. Could anyone shed some light?
onimoni
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