quiz_03_laqf_07_solution

# quiz_03_laqf_07_solution - dimension Alternatively you can...

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Linear Algebra, Fall 2007 Quiz 3 hints 1. Consider the functions 2 x and | x | . (a) Are they linearly independent in C [0 , 1]? Explain. Obviously 1 · 2 x - 2 · | x | = 0 on [0 , 1], so they are linearly dependent. (b) Are they linearly independent in C [ - 1 , 1]? Explain. Check out the determinant formed by evaluating the two functions at 1 and - 1. They are linearly independent. When you suspect some functions are linearly dependent, try to ﬁnd a non-trivial coeﬃcient vector ( (1 , - 2) in (a) ). When you think they are linearly independent, try to evaluate the functions at some points (for example x = 1 and - 1 here) to form a non-singular matrix. Or take successive derivatives at one point to form a non-singular matrix (the Wronskian). 2. Find the dimension of the subspace of P 9 spanned by x , 2 x +1, x - 1, x 2 +1 and x 2 - 1. There is a standard procedure to reduce the spanning set by one vector at a time, keeping the span unchanged. Do it until they are linearly independent, this gives the
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Unformatted text preview: dimension. Alternatively, you can do the following: see if you can generate 1 from these polynomi-als, then try to generate x and x 2 , etc. until it is obviously impossible ( x 3 is impossible here, for example). Then count how many you have generated. 3. Let E = { u 1 , u 2 } , F = { v 1 , v 2 } be two bases in R 2 . where u 1 = ± 1 2 ² , u 2 = ± 3 4 ² , v 1 = ± 4 3 ² , v 2 = ± 2 1 ² . If x = 5 u 1 + 6 u 2 , ﬁnd the coordinate of x in F . Verify your answer in the end. Most people did this right. If not, read the book. 4. Let x ∈ R m , y ∈ R n be nonzero vectors in R m and R n , respectively, and let A = xy T . What is the dimension of N ( A )? Explain. Try with some actual vectors x and y , see if you can read the row space and the column space. It is easy to see the rank is 1 and one can proceed to ﬁnd dimension of N ( A ) by a Theorem in the book....
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