Linear Algebra with Applications (3rd Edition)

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ma253 , Fall 2007 – Midterm 1 Solutions In each of the ten questions that follow, decide whether the statement being made is true or false. Indicate your decision by writing T or F in the space provided. Each question is worth four points. 1. If a subspace V of R 4 does not contain any of the standard basis vectors e 1 , e 2 , e 3 , e 4 , then it must consist only of the zero vector. This is false . Consider the next-simplest kind of subspace, the span of one vector, i.e., a line through the origin in R 4 . There’s no reason such a line would have to contain one of the basis vectors. 2. There exists a system of three linear equations in three unknowns that has exactly two solutions. This is false , as we discussed in class. The only options for linear spaces are a unique solution, no solutions, or infinitely many solutions. 3. If the vector u is a linear combination of the vectors v and w , then Span ( u, v, w, y ) = Span ( v, w, y ) . This is true . We are told that u = αv + βw for some α and β . That allows us to substitute for u wherever it appears an expression involving only v and w . In other words, we can rewrite any linear combination of u , v , w and y as a linear combination involving only v , w and y . So the spans, i.e., the sets of all possible linear combinations, are the same. 4. The only 2 × 2 matrices A such that A 2 = I are 1 0 0 1 and - 1 0 0 - 1 . Very false . For example, - 1 0 0 1 , 1 0 0 - 1 , 0 1 1 0 , 3 5 4 5 4 5 - 3 5 all have that property. In fact, any reflection has that property. Just think of what it says in transformation terms: that doing the transformation twice gives the identity transfor- mation. 5. If A and B are n × n matrices, then ( A + B ) 2 = A 2 + 2AB + B 2 . This is false . Of course, if A and B happen to commute, then it would be true for those particular matrices A and B . But it is false in general. The object of opening the mind, as of opening the mouth, is to shut it again on something solid. – G. K. Chesterton
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6. Let P 5 be the linear space of polynomials of degree less than or equal to 5 . Define a function T from P 5 to P 5 by T ( p ) = p - 3p + 2p . Then T is a linear transformation. True , because of standard properties of derivatives. 7. The image of a 4 × 3 matrix is a subspace of R 4 . True , since such a matrix represents a transformation R 3 - R 4 . 8. Let W be the set of vectors x 1 x 2 x 3 in R 3 such that x 1 + x 2 + x 3 = 0 . Then W is a subspace of R 3 . True . The artsy-fartsy way of seeing it is to note that sending x 1 x 2 x 3 to x 1 + x 2 + x 3 is a linear transformation from R 3 to R 1 and W is its kernel. 9. If A and B are matrices, then rref ( AB ) = rref ( A ) rref ( B ) . False , but fairly tricky, because it’s true for invertible A and B (remember that for invertible matrices the rref is always just the identity matrix). So you had to either go look for the right counterexample or think about what this is claiming about the elementary matrices (it ends up saying that they commute with any matrix).
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