Consider the following subset of R^2: W = {(y1 y2) in R^2 such that y1 = x_1 + 3
ID: 3036781 • Letter: C
Question
Consider the following subset of R^2: W = {(y1 y2) in R^2 such that y1 = x_1 + 3x_2 + 6x_3 and y_2 = x_1 + 2x_2 + 5x_3 for some vector (x_1 x_2 x_3) in R^3}. Write W as image of a matrix A. In this way W becomes a subspace as the image of a matrix is a subspace. Find a basis of W and its dimension. Use then the Fundamental Theorem of Linear Algebra to find the dimension of the kernel of A. Answer True or False to the following statements. You may want to review Theorem 3.3.4 in your textbook. (i). If three vectors v_1, v_2, v_3 of R^3 span R^3, then the span of v_1, v_2, v_3, w, where w is any vector of R^3, is again R^3. (ii). If two vectors of R^4 span a plane through the origin of R^4, then they are linearly independent. (iii). The span of 4 linearly independent vectors of R^7 is of dimension 4. (iv). The span of any 4 vectors in R^7 is of dimension 4. (v). Consider two vectors v_1, v_2 of R^3 that span a plane through the origin. Then it is possible to find a third vector v_3 of R^3 such that v_1, v_2, v_3 form a basis of R^3 (vi). Consider a plane through the origin in R^10 Then a basis of the plane is also a basis of R^10. (vii). The dimension of the span of m vectors in R^n is at most m. Moreover it is equal to m if the vectors are linearly independent.Explanation / Answer
(i) false it will not span of R3
(ii) true the two vectors are linearly independent
(iii)true span of 4 linearly independent vector of R7 is of dimension 4
(iv) false the span of any 4 vectors in R7 is of dimension 4
(ii)