1.Calculate A
1000for the matrix A defined in Holt, Section 6.4, problem 20, by the following process:
Show that A can be diagonalized by finding an invertible matrix P and a diagonal matrix D such that A = PDP
-1, and then calculate (PDP
-1)
1000.
2. Suppose that A is a diagonalizable matrix with distinct nonzero eigenvalues. Prove that A^2 has positive eigenvalues.
3. find the eigenvalues and a basis for each eigenspace for the given matrix:
A= [[1,-2][1,3]]
4. A vector space V and a subset S is given. Determine if S is a subspace of V, and if so, prove it. If not, give an example showing one of the conditions of definition 7.3 is not satisfied.
Def 7.3: a subset S of a vector space V is a subspace if S satisfied following 3 conditions
- S contains 0 the zero vector
- If u and v are in s, then u + v is also in s
- If c is a scalar and v is in s, then cv is also in s
V=C[-3,3] and S is the set of real valued functions f such that f(-1) + f(1) =0.
5. (same instructions as #4).
V=T(3,3) and S is the set of invertible linear transformations.
6. determine if the vector is in the subspace of R(2X3) given by SPAN {[[1,2,1][0,1,3]] , [[0,3,1][-1,1,0]]}
V= [[2,1,1][1,1,5]]
7. State whether following statement is true or false. Justify your answer.
if {v1,v2,v3} is a linearly independent set, then so is {v1, v2-v1, v3-v2+v1}
8. determine if the set
Vis a basis for V
V= {x^2-5x+3,3x^2-7x+5,x^2-x+1},
V=P^2
9. Find a basis for the subspace S and determine dim(s).
S is the subspace of P consisting of polynomials p such that p(0)=0.
10. Note:w Suppose that u1 and u2 are orthogonal vectors, with ||u1|| =3 and ||u2||=4. Find ||2u1-u2||.
11. Find a basis for S - for the subspace S.
S=SPAN [[1][1][-2]]
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1. Calculate A1000 for the matrix A defined in Holt, Section 6.4, problem 20, by the following process: Show that A can be diagonalized by finding an invertible matrix P and a diagonal matrix D such that A = PDP-1, and then calculate (PDP-1)1000. 2. Suppose that A is a diagonalizable matrix with distinct nonzero eigenvalues. Prove that A^2 has positive eigenvalues. 3. find the eigenvalues and a basis for each eigenspace for the given matrix: A= [[1,-2][1,3]] 4. A vector space V and a subset S is given. Determine if S is a subspace of V, and if so, prove it. If not, give an example showing one of the conditions of definition 7.3 is not satisfied. Def 7.3: a subset S of a vector space V is a subspace if S satisfied following 3 conditions S contains 0 the zero vector If u and v are in s, then u + v is also in s If c is a scalar and v is in s, then cv is also in s V=C[-3,3] and S is the set of real valued functions f such that f(-1) + f(1) =0. 5. (same instructions as #4). V=T(3,3) and S is the set of invertible linear transformations. 6. determine if the vector is in the subspace of R(2X3) given by SPAN {[[1,2,1][0,1,3]] , [[0,3,1][-1,1,0]]} V= [[2,1,1][1,1,5]] 7. State whether following statement is true or false. Justify your answer. if {v1,v2,v3} is a linearly independent set, then so is {v1, v2-v1, v3-v2+v1} 8. determine if the set V is a basis for V V= {x^2-5x+3,3x^2-7x+5,x^2-x+1}, V=P^2 9. Find a basis for the subspace S and determine dim(s). S is the subspace of P consisting of polynomials p such that p(0)=0. 10. Note:w Suppose that u1 and u2 are orthogonal vectors, with ||u1|| =3 and ||u2||=4. Find ||2u1-u2||. 11. Find a basis for S - for the subspace S. S=SPAN [[1][1][-2]]