Consider the following three vectors in R: V = (a) Show that = (V, V2, V3)...

 

Transcribed Image Text:

Consider the following three vectors in R³: V₁ = (a) Show that = (V₁, V2, V3) is a basis for R³. Consider the following four vectors in R4: = 1 --0--0---0--(-)) = 1 -(). »-(-9). »-0). 0 = V3 = 3 1 1 (b) Show that B = (U₁, U2, U3, U4) is a basis for R¹. A linear map T: R¹4 → R³ is determined by: (d) Give a basis for the image of T. = (e) Give a basis for the kernel of T. T(u₁)= V₁, T(u₂) = 2v2, T(u3) = T(U4) = 0, (c) Give the standard matrix of T, i.e., in terms of the standard bases for R4 and R³. = For all the points above, write full sentences explaining all the steps. 1 [4 Marks] [4 Marks] [9 Marks] [4 Marks] [4 Marks] Consider the following three vectors in R³: V₁ = (a) Show that = (V₁, V2, V3) is a basis for R³. Consider the following four vectors in R4: = 1 --0--0---0--(-)) = 1 -(). »-(-9). »-0). 0 = V3 = 3 1 1 (b) Show that B = (U₁, U2, U3, U4) is a basis for R¹. A linear map T: R¹4 → R³ is determined by: (d) Give a basis for the image of T. = (e) Give a basis for the kernel of T. T(u₁)= V₁, T(u₂) = 2v2, T(u3) = T(U4) = 0, (c) Give the standard matrix of T, i.e., in terms of the standard bases for R4 and R³. = For all the points above, write full sentences explaining all the steps. 1 [4 Marks] [4 Marks] [9 Marks] [4 Marks] [4 Marks]