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Unformatted text preview: MATH 33A LECTURE 2 15T MIDTERM 2 Problem 1. (True/ False, 2 pts each) Mark your answers by filling in the appropriate box
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(iii If A iS an invertible n X 11 matrix, then the rank of A is 11. (iii: There exists a matrix A for which ker A is the same as the image of A.
(iVI If AB 2 BA for two 11 x n matrices A, B, then either A or B must be the identity 2 3 7 1 O —1
(i! The reduced row echelon form of the matrix [1 4 8] is [0 1 O matrix.
1 O —1
(V3 Asssume that the reduced row echelon form of A is O 1 O . Does the sys—
0 O O tem of linear equations A56 = 0 have more than one solution?
(Vii If A  A  A is the identity matrix, then A is invertible.
(vii: The columns of a matrix A always form a basis for the image of A.
(Viﬁi If T and S are two linear transformations whose matrices [T] and [S] are equal,
then T = 8.
(ix: LEt A be an n x 111 matrix and B and m X 71 matrix (here it may be that n aé m) If A is not invertible, is it true that also AB is not invertible?
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independent? (b) Find a basis for the image of A. (c) Find the kernel of A. 7 1
8 O . (a) Are the columns of A linearly
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Problem 3. (20 pts) Let L be the line spanned by the vector 2 . (a) Find a matrix A so
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that L is the image Of A 0’) Find a matrix B so that L is the kernel of B.
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1 MATH 33A LECTURE 2 IST MIDTERM 6 Problem 5 (20.1%) Let A be an n X 111 matrix, and let B be an invertible m X m matrix. (f1)
Show that the Image of A is the same as the image of AB. (b) Express the rank of AB In
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 Fall '09
 DAI,S.
 Linear Algebra, Row echelon form, MATH 33A LECTURE, VCQQTW

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