2. (6 points) Let us consider the vector of colors produced by a color copier. A

Question

2. (6 points) Let us consider the vector of colors produced by a color copier. Adding two colors means combining the two colors on paper. Multiplying by a constant means changing the intensity (or darkness) of the color. Although technically, the set of all colors is not a vector space, consider the following. (a) Is the set of colors (blue, green, yellow linearly independent? Explain. (b) Is the set of colors (blue, yellow, red) linearly independent? Explain. 3. (9 points) For the following, be sure to justify your answer. (a) (3 points) How many pivot columns must a 5 × 4 matrix have if its columns are linearly (b) (3 points) How many pivot columns must a 4 × 6 matrix have if its columns span R (c) (3 points) Let A be a 4 × 5 matrix. Can the columns of A be linearly independent? independent? Justify your an Justify your answer. Explain

Explanation / Answer

3(a)

A 5,4 matrix means a matrix with 5 rows and 4 columns .We know that number of lineary independent row and number of linearly independent columns of a matrix are same.

Since given that columns are linearly independent (there are 4 linearly independent columns ) ,hence we will have only 4 pivots coloumns in the matrix .

(b)

Given that there is 4,6 matrix and its columns span R^4 .Since the dimension of R^4 is 4 ,4 columns of the matrix are linearly independent (rank of the atrix is 4) ,hence there are 4 pivots columns in the matrix .

(c)

Let A be 4,5 matrix .Then rank (A) <=min(4,5)=4

that is rank (A) <=4.

number of linearly independent rows of A = number of linearly columns of A <=4

Since nummber of columns of matrix A is 5 ,columns of A can not be linearly independent .

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