lin alg

Subspace test for first quadrant V={x≥0,y≥0}

Check closure: u+v stays in V

Counterexample for V not subspace

Pick u=[1,1], c=-1 ⇒ cu=[-1,-1] not in V.

Subspace test for W={xy≥0}

Scalar mult OK

Polynomials p(t)=at² subspace test

Closed under addition & scalar mult → subspace.

Polynomials p(t)=a+t² subspace test

Scalar mult breaks constant term → not a subspace.

Polynomials deg≤3 with integer coeffs subspace test

Not closed under real scalar multiplication → not a vector space.

Set p(0)=0 subspace test

Sums and scalar multiples still satisfy p(0)=0 → subspace.

Find v so { [s,3s,2s] } = Span{v}

Factor: s[1,3,2] → v=[1,3,2].

Show H={ [2t,0,-t] } is subspace

Factor: t[2,0,-1] → H=Span{[2,0,-1]}.

Functions y=c₁cos(ωt)+c₂sin(ωt) form subspace

Linear combination keeps same form → subspace.

H={A: FA=0} subspace test

Kernel of a linear transformation → always a subspace.

Test w∈Nul A

Compute A w = 0?

Find basis for Nul A

RREF A → solve Ax=0 → express solution with free vars → basis vectors.

Find A s.t. given set is Col A

The spanning vectors ARE the columns.

Find nonzero vectors in Nul A, Col A, Row A

Nul from solving Ax=0

Nul A = kernel T(x)=Ax (T/F)

True.

Determine if set is basis for R³

Check 3 vectors, linear independence ⇔ determinant≠0.

Test basis when a vector is zero

Any zero vector → automatically not independent → not basis.

Find if set spans R³

Check rank = 3 (or pivot in every column of 3×3 matrix).

Find basis for Nul of given matrix

RREF → free vars → parametric vector form → basis.

Find basis for plane x+4y−5z=0

Solve as homogeneous system → express free vars → two basis vectors.

Find Nul A, Col A, Row A

Nul from RREF

Show w∈Span{v1,v2,v3}

Solve c1v1+c2v2+c3v3=w.

Test if y∈Span(columns of A)

Solve Ax=y

Find x from [x]_B

x = B * [x]_B (linear combo of basis vectors).

Find change of basis matrix B→C

Columns = coordinates of b1,b2 in basis C.

Coordinate of polynomial in basis B

Solve c1b1+c2b2+c3b3 = p(t).

Coordinate map 1-1 (T/F)

True.

Coordinate map onto R^n (T/F)

True.

Test linear independence of polynomials using coordinate vectors

Convert each polynomial to coord vector, check if independent.

Find basis of subspace described by equations

Solve system → parametric vector form → basis.

Find dimension of subspace spanned by vectors

Row-reduce matrix with vectors as rows or columns → rank = dimension.

Dimensions of Nul A, Col A, Row A

dim Col = pivot columns

Hermite polynomials form basis for P₃

4 polynomials, independent → basis.

Find coordinates of p(t) in Hermite basis

Solve c1H1+c2H2+c3H3+c4H4 = p(t).

Change-of-coordinates matrix B→C using relations

Write b’s in terms of c’s → columns of matrix.

Compute [x]_C given x in B

[x]C = (change-of-basis matrix) * [x]B.

Change-of-coordinates F→D

Express f’s in terms of d’s → columns.

Which equation holds for change-of-basis matrix P?

[x]W = P [x]U.

Find change-of-basis matrix from polynomial basis to standard

Expand each basis poly → coefficients become columns.

Write t² in basis B

Solve c1b1+c2b2+c3b3 = t².

Check if λ is eigenvalue

Solve det(A−λI)=0.

Check if vector is eigenvector

Evaluate A v = λv? If scalar multiple yes → eigenvalue λ.

Find basis for eigenspace

Solve (A−λI)x=0 → basis of null space.

Find eigenvalues of triangular matrix

Eigenvalues are diagonal entries.

Eigenvalues of A⁻¹

λ eigenvalue of A → λ⁻¹ eigenvalue of A⁻¹.

If A²=0 eigenvalues

Only eigenvalue = 0.

Find characteristic polynomial 2×2

det(A−λI)= (a−λ)(d−λ)−bc.

Find eigenvectors 2×2

Plug eigenvalue into (A−λI)x=0.

Characteristic polynomial 3×3

Expand using cofactor expansion.

Eigenvalues from upper triangular block matrix

Diagonal entries (repeated as needed).

Compute Aᵏ using A=PDP⁻¹

Aᵏ = P Dᵏ P⁻¹.

Diagonalize matrix

Find eigenvalues, find eigenspaces, check dimension sum = n.