Question

(a) Let A be a 4 × 5 matrix with rank(A) = 3. Find dim(col(A)). (b) Let A be a 5 × 6 matrix with dim(null(A)) = 2. Find dim(row(A)). (c) Let A be a 6 × 7 matrix with dim(row(A)) = 5. Find dim(null(A)). (d) Let A be a 4 × 6 matrix with rank(A) = 3. Find dim(null(A)). (e) Let A be a 6 × 8 matrix. What is the smallest value dim(null(A)) can have? 

          (a) Let A be a 4 × 5 matrix with rank(A) = 3. Find dim(col(A)).
(b) Let A be a 5 × 6 matrix with dim(null(A)) = 2. Find dim(row(A)).
(c) Let A be a 6 × 7 matrix with dim(row(A)) = 5. Find dim(null(A)).
(d) Let A be a 4 × 6 matrix with rank(A) = 3. Find dim(null(A)).
(e) Let A be a 6 × 8 matrix. What is the smallest value dim(null(A)) can have?
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(a) Let A be a 4 times; 5 matrix with rank(A) = 3. Find dim(col(A)). (b) Let A be a 5 times; 6 matrix with dim(null(A)) = 2. Find dim(row(A)). (c) Let A be a 6 times; 7 matrix with dim(row(A)) = 5. Find dim(null(A)). (d) Let A be a 4 times; 6 matrix with rank(A) = 3. Find dim(null(A)). (e) Let A be a 6 times; 8 matrix. What is the smallest value dim(null(A)) can have?

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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(a) Let A be a 4 × 5 matrix with rank(A) = 3. Find dim(col(A)). (b) Let A be a 5 × 6 matrix with dim(null(A)) = 2. Find dim(row(A)). (c) Let A be a 6 × 7 matrix with dim(row(A)) = 5. Find dim(null(A)). (d) Let A be a 4 × 6 matrix with rank(A) = 3. Find dim(null(A)). (e) Let A be a 6 × 8 matrix. What is the smallest value dim(null(A)) can have?
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Transcript

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00:01 In a part of the question, we are given a matrix a.
00:07 And the order of matrix is given us 4 cross 5.
00:12 Also we are given rank of matrix a is equals to 3.
00:18 And we have to find dimension of column space a.
00:22 So as we know, rank of a must be equal to dimension of column space.
00:34 Of a.
00:35 So from here we have dimension of column space of a is equals to 3 as the rank of matrix a is 3.
00:46 So this is the final answer for a part of the question.
00:50 Now moving towards the b part of this question here we are given matrix a.
00:58 The order of this matrix is 5 cross 6.
01:02 Here dimension of null space of of a is given as 2 and we have to find dimension of row space of 2.
01:14 So as we know dimension of row space sorry row space of a is equals to number of columns minus dimension of null space of a.
01:35 This is equals to number of columns are 6 minus dimension of null space of a.
01:42 Of a is 2.
01:43 6 minus 2 is 4 so from here we can say dimension of row space of a is equals to 4 and this is the final answer for b part of the question...
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