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# q3ts - A is also a basis for the image 2 Find a matrix A so...

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Quiz III 1. Find a basis for the image of the transformation T ( ⃗x ) = A⃗x , when A = 0 1 1 4 0 0 1 3 0 1 0 1 Solution: The ﬁrst column is the zero vector which can never be part of a linearly independent set of vectors. The fourth column is the sum of the second column and three times the third column. The second and third columns are linearly independent because the third column has a 1 in the row where the second column has a 0 (and vice versa). So, since the image is the set of linear combinations of the columns, { 1 0 1 , 1 1 0 } is a basis for it. That is the set that you would get by removing “redundant” vectors, but any pair of columns chosen from the last three columns of
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Unformatted text preview: A is also a basis for the image. 2. Find a matrix A so that the image of the transformation T ( ⃗x ) = A⃗x from R 2 to R 3 is the plane through the origin with normal vector (1 , 1 , 1). Solution. That plane is the set of linear combinations of (1 , ,-1) and (1 ,-1 , 0), since they are both perpendicular to (1 , 1 , 1) and are linearly independent. So A = 1 1-1-1 will work. There are, of course, many other choices. You can take the columns of A to be any pair of linearly independent vectors that are perpendicular to (1 , 1 , 1)....
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