LLM Primer

Solutions

MATH-C05-C-001

Add the equations:

(x+y)+(x−y)=7+3(x + y) + (x - y) = 7 + 3

So:

2x=102x = 10

Therefore x = 5.

MATH-C05-C-002

2v1+v2=2[1,2]+[3,0]2v_1 + v_2 = 2[1, 2] + [3, 0]

So:

[2,4]+[3,0]=[5,4][2, 4] + [3, 0] = [5, 4]

MATH-C05-C-003

No.

The set spans the plane, but it is redundant because:

[1,1]=[1,0]+[0,1][1, 1] = [1, 0] + [0, 1]

A basis must span the space and avoid redundant directions.

MATH-C05-C-004

The columns [2, 0] and [4, 0] point along the same line.

There is only one independent output direction, so the rank is 1.

MATH-C05-C-005

Yes.

The column space is the set of all combinations of [2, 0] and [4, 0]. Those combinations lie on the horizontal line. The vector [6, 0] is on that line.

For example:

3[2,0]+0[4,0]=[6,0]3[2, 0] + 0[4, 0] = [6, 0]

MATH-C05-C-006

Yes.

For [-2, 1]:

2(−2)+4(1)=−4+4=02(-2) + 4(1) = -4 + 4 = 0

So the matrix maps [-2, 1] to [0, 0].

MATH-C05-C-007

A[1,0]=[4,0]=4[1,0]A[1, 0] = [4, 0] = 4[1, 0]

The eigenvalue is 4.

MATH-C05-C-008

P^{-1} acts first.

In A = PDP^{-1}, the rightmost factor touches the input vector first. It changes the vector into the eigenvector coordinate system.

MATH-C05-C-009

A rank-2 approximation keeps the two largest singular values.

From 8, 3, 0.5, and 0.1, it keeps:

8,38,\quad 3

MATH-C05-C-010

No.

The first principal component is the direction of largest variance. Variance is not the same as causality, fairness, or predictive usefulness.

MATH-C05-C-011

Since x_0 is a solution:

Ax0=bAx_0 = b

Since z is in the null space:

Az=0Az = 0

Therefore:

A(x0+z)=Ax0+Az=b+0=bA(x_0 + z) = Ax_0 + Az = b + 0 = b

So x_0 + z is also a solution.

MATH-C05-C-012

A 3 x 3 matrix starts with a three-dimensional input space.

Rank 2 means only two independent output directions remain visible. At least one independent input direction is collapsed or made redundant by the matrix.

MATH-C05-C-013

Singular values measure strengths of directions.

A singular value of 0 means the matrix carries no output strength in that direction. That direction is collapsed.

MATH-C05-C-014

A rank-2 approximation keeps the two strongest singular directions.

That can be useful compression, but it can still remove small directions that matter for a task, especially rare details or weak signals.