MATH 46, REVIEW FOR FINAL

1. (a) Let A∈Mm×n,A= [aij ]. Deﬁne the following terms: cofactor of aij , det A,eigenvalue

of A,eigenvector associated to an eigenvalue of A, eigenspace associated to eigenvalue of

A,Ais diagonalizable.

(b) For a function T:V→Vdeﬁne the following terms: Tis a linear transformation,Tis

diagonalizable.

(c) Describe the least squares method to solve Ax=b. When might you use it to solve

Ax=b?

(d) What deﬁnitions do you think will be on the test?

2. Let A∈Mm×n. Assume for all x∈Rnthat Ax=0. Prove that Ais the zero matrix.

3. Solve the following linear system:

2x+y−2z= 10

3x+ 2y+ 2z= 1

5x+ 4y+ 3z= 4

(a) by row reduction.

(b) by Cramer’s Rule. [NOT COVERED IN 2018]

4. Let Abe an m×nmatrix and let TA:Rn→Rmbe deﬁned by TA(x) = Ax. Are the following

statements true or false. If true give a proof, if false explain why.

(a) dim Nul A≤n.

(b) rank A≤m.

(c) If n > m then the linear transformation TAcannot be one-to-one.

(d) If n < m then TAcannot be onto.

(e) If TAis one-to-one and m=nthen TAmust be onto.

5. Let Abe an n×nmatrix satisfying A3=In, where Inis the n×nidentity matrix. Show that

det A= 1.

6. For the following problems, prove the statement or give a speciﬁc counterexample:

(a) W={p∈P2

(p(3))2+p(3) = 0}is a subspace of P3.

(b) W={(x, y)∈R2

x+y= 0}is a subspace of R2.

(c) Let Vbe a vector space and let f1:V→Rand f2:V→Rbe linear transformations.

Deﬁne T:V→R2deﬁned by T(v) = (f1(v), f2(v)). Tis linear.

7. Let S=2 1

0 1 ,1 0

1 1 ,8 2

4 6 .

(a) Decide whether Sis independent.

(b) Let W= span S. Use the result of (a) to ﬁnd a basis of Wthat is a subset of S. Find

dim W.

(c) Determine whether 5 1

−1 9 ∈W= span S.

8. If you are given a square upper triangular matrix, how would you tell at a glance whether or

not it is invertible? Explain your answer using determinants. How would you tell at a glance

the eigenvalues of the matrix and their multiplicities?

9. Let Vand Wbe vector spaces and let T:V→Wbe a linear transformation.

(a) Let S={v1,v2,v3,...,vℓ}be a set of vectors that spans V. Prove that the set

{T(v1), T (v2), T (v3), . . . , T (vℓ)}spans range Tin W.

(b) Assume Tis one-to-one, and assume Sis a basis of V. Prove the set

{T(v1), T (v2), T (v3), . . . , T (vℓ)}is independent.

(c) Show that L:P3→R3be deﬁned by L(p) = (p(0), p(1), p(3)). Prove Lis a linear

transformation.

(d) Use the result of (a) and the basis B={1, t, t2, t3}of P3to ﬁnd a spanning set of range L.

Find a subset of this spanning set that is a basis of range L.

1

## Document Summary

Math 46, review for final: (a) let a mm n, a = [aij]. De ne the following terms: cofactor of aij, det a, eigenvalue of a, eigenvector associated to an eigenvalue of a, eigenspace associated to eigenvalue of. A, a is diagonalizable. (b) for a function t : v v de ne the following terms: t is a linear transformation, t is diagonalizable. (c) describe the least squares method to solve ax = b. Ax = b? (d) what de nitions do you think will be on the test: let a mm n. Assume for all x rn that ax = 0. Prove that a is the zero matrix: solve the following linear system: 5x + 4y + 3z = 4 (a) by row reduction. (b) by cramer"s rule. [not covered in 2018: let a be an m n matrix and let ta : rn rm be de ned by ta(x) = ax.