math 235 ch 1 practice

math 235 ch 1 practice - rank of the augmented matrix is 4...

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Math 235 - Section 6, Chapter 1 Practice Problems Adam Gamzon 1. Using Gauss-Jordan elimination solve the following linear systems and write your solutions (if there are any) in vector form: (a) 3 x + 11 y + 19 z = - 2 , 7 x + 23 y + 39 z = 10 , - 4 x - 3 y - 2 z = 6 . (b) x 1 - 7 x 2 + 5 x 5 = 3 , x 3 - 2 x 5 = 2 , x 4 + x 5 = 1 . 2. Find the polynomial f ( t ) of degree 3 such that f (1) = 1, f (2) = 5, f 0 (1) = 2 and f 0 (2) = 9. Hint: write f ( t ) = at 3 + bt 2 + ct + d and try to solve a linear system where a, b, c, and d are the variables. 3. Compute the rank of 1 4 7 2 5 8 3 6 9 . 4. For which values of the constants b and c is the vector 3 b c a linear combination of 1 3 2 , 2 6 4 , and - 1 - 3 - 2 ? 5. (a) Consider a linear system Ax = b where A is a 4 × 3 matrix. If rank( A | b ) = 4 (that is, the
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Unformatted text preview: rank of the augmented matrix is 4), how many solutions does this system have? (b) More generally, show that a linear system Ax = b is consistent if and only if rank A = rank( A | b ). Hint: remember that for “if and only if” statements, you have to show both directions - if Ax = b is consistent then rank A = rank( A | b ) and if rank A = rank( A | b ) then Ax = b is consistent. 6. Prove that if a vector v in R 4 is a linear combination of the vectors u and w in R 4 and if A is a 5 × 4 matrix then Av is a linear combination of Au and of Aw . 1...
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