1. (12 points) For these problems, begin with what is given (mathematically) and
ID: 3169311 • Letter: 1
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1. (12 points) For these problems, begin with what is given (mathematically) and then use math- ematical properties (such as those for matrix-vector products) to get the result desired. (a) (6 points) Consider the system of equations given by Ax. Suppose v and w are solutions. Show that v w is also a solution, i.e. show that A() 0 (b) (6 points) Suppose Au-0 and Aw = b where b0. Show that for any scalar (constant) 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)lnearly 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 x 4 matrix have if its columns are linearly independent? Justify your answer Justify your answer Explain. (b) (3 points) How many pivot columns must a 4 x 6 matrix have if its columns span R1? (c) (3 points) Let A be a 4 x 5 matrix. Can the columns of A be linearly independent? 4. (9 points) No explanation necessary for these questions. (a) Give an example of a set of vectors in R3 that are linearly dependent and span R3 (b) Give an example of a set of vectors in R3 that are linearly independent and do not span IR (c) Give an example of a set of vectors inR that are linearly independent and span RExplanation / Answer
(b).If Av = 0 and Aw = b(b0), then for any scalar c, we have A(cv+w) = A(cv)+Aw = cAv+Aw = c.0+b = 0+b = b.
2. (a). The colr green is a combination nof blue and yellow colors so that the set {blue,green,yellow} is not linearly independent.
(b).The set {blue,yellow, red} is linearly independent as none of these colors is a comniation of the other 2 colors.
3. (a).All the 4 columns of a 5x4 matrix must be pivot columns if the columns are to be linearly independent otherwise at least one of its columns will be a linear combination of the remaining columns.
(b).A 4x6 matrix has 6 columns , eachof which is a 4-vector. Since the dimension of R4 is 4, hence of the columns of a 4x6 matrix must be pivot columns so that its columns span R4.
(c ). A 4x5 matrix has 5 columns , eachof which is a 4-vector. Since the dimension of R4 is 4, hence of the columns of a 4x5 matrix cannot be linearly independent.
4. (a). The vectors u= (1,0,0)T ,v = (0,1,0)T , w = (0,0 ,1)T and z = (1,1,1)T are linearly dependent as z = u+v+w. Further,the set {u,v ,w,z} spans R3 as any vector in R3 can be expressed as a linear combination of these vectors.
(b). The vectors u= (1,0,0)T and v = (0,1,0)T are linearly independent, but do not span R3 as the vector (0,0 ,1)T caannt be expressed as a linear combination of these vectors.
(c ). The vectors u= (1,0,0)T ,v = (0,1,0)T and w = (0,0 ,1)T are linearly independent. Further,the set {u,v ,w} spans R3 as any vector in R3 can be expressed as a linear combination of these vectors.
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