Assignment5
(a) (5 points) Find a basis fu1;u2;u3g for IR3, such that P is the change of coordinates matrix from
fu1;u2;u3g to the fv1;v2;v3g. Hint: What do the columns of P
represent?
(b) (5 points) Find a basis w1;w2;w3 for IR3, such that P is the change of coordinates matrix from
fv1;v2;v3g to fw1;w2;w3g.
2. Let = fb1;b2g and C = fc1;c2g be bases for IR2. In the following subparts nd the change of
coordinates matrix from to C. Also nd the change of coordinates matrix from C to .
(a) (3 points) b1 =
3. (5 points) In IP2 nd the change of coordinate matrix from the basis =f1 2t+t2;3 5t+4t2;2t+3t2g
to the standard basis C =f1;t;t2g. Then nd the coordinate vector for 1 + 2t.
4. (5 points) Determine whether w is in the column space of A, the null space of A or both, where:
5. Determine whether the sets of polynomials form. a basis for IP3. Justify your conclusions:
(a) (3 points) 3 + 7t;5 +t 2t3;t 2t2;1 + 16t 6t2 + 2t3
(b) (3 points) 5 3t+ 4t2 + 2t3;9 +t+ 8t2 6t3;6 2t+ 5t2;t3
All matrices are in capital letters and bold. All vectors are in lower case and bold. All scalars are lower case and not
bolded.
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6. (5 points) Let S be a nite set in a vector space V with the property that every x in V has a unique
representation as a linear combination of elements of S. Show that S is a basis of V.
7. (5 points) Let = fb1;:::;bng be a basis for the vector space V. Explain why the coordinates of
coordinate vectors b1;:::;bn are the columns e1;:::;en of the nxn identity matrix. Note that e1;:::;en
are the standard basis.
8. (5 points) Compute determinant of B4, where B =
9. Let A and B be a 3 x 3 matrices with detA = 3 and detB = 4. Use properties of determinant and
nd the following:
(a) (2 points) det AB
(b) (2 points) det 5A
(c) (2 points) det BT
(d) (2 points) det A 1
(e) (2 points) det A3
10. Find the determinant of the following when you know that: det
(b) (3 points)
11. Use cofactor expansion to nd the determinant of the following matrices. Make sure to clearly tell us
which row/column you have chosen for the expansion.
(a) (3 points)
12. Suppose memory or size restrictions prevent your matrix program from working with matrices having
more than 32 rows and 32 columns and suppose some project involves 50x50 matrices A and B.
(a) (5 points) Solve Ax = b for some vector b in IR50, assuming that A can be partitioned into a 2x2
block matrix Aij, with A11 an invertible 20 x 20 matrix, A22 an invertible 30 x 30 matrix, and
A12 a zero matrix. Hint: Describe appropriate smaller systems to solve, without using any matrix
inverse.