field. For any A € M₂(K), let Let I2 € M₂(K) be the 2 × 2 identity matrix, where K is a (a) Let K = R and A V₁ = I2, V₂ = A, and regard them as elements of V = = 01 V3 = A² M₂(K) as a vector space. (i) Show that v₁, V2, V3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(x). Justify your answers.
field. For any A € M₂(K), let Let I2 € M₂(K) be the 2 × 2 identity matrix, where K is a (a) Let K = R and A V₁ = I2, V₂ = A, and regard them as elements of V = = 01 V3 = A² M₂(K) as a vector space. (i) Show that v₁, V2, V3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(x). Justify your answers.
field. For any A € M₂(K), let Let I2 € M₂(K) be the 2 × 2 identity matrix, where K is a (a) Let K = R and A V₁ = I2, V₂ = A, and regard them as elements of V = = 01 V3 = A² M₂(K) as a vector space. (i) Show that v₁, V2, V3 are linearly dependent as elements of V. (ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3? (b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are linearly independent? You may wish to consider the characteristic polynomial PA(x). Justify your answers.
Let I2 ∈ M2(K) be the 2 × 2 identity matrix, where K is a field. For any A ∈ M2(K), let
v1 =I2, v2 =A, v3 =A^2 and regard them as elements of V = M2(K) as a vector space.
(a) Let K=R and A=| 1 1, 0 1| (2x2 matrix)
(i) Show that v1, v2, v3 are linearly dependent as elements of V
(ii) What is the dimension of the vector space ⟨v1, v2, v3⟩ spanned by v1, v2, v3?
(b) Does there exist a choice of field K and of matrix A such that v1,v2,v3 are linearly independent? You may wish to consider the characteristic polynomial pA (x).
Justify your answers.
Transcribed Image Text:field. For any A € M₂(K), let
Let I2 € M₂(K) be the 2 × 2 identity matrix, where K is a
(a) Let K = R and A
V₁ = I2, V₂ = A,
and regard them as elements of V
=
= 01
V3 = A²
M₂(K) as a vector space.
(i) Show that v₁, V2, V3 are linearly dependent as elements of V.
(ii) What is the dimension of the vector space (v₁, V2, V3) spanned by V₁, V2, V3?
(b) Does there exist a choice of field K and of matrix A such that V₁, V2, V3 are
linearly independent? You may wish to consider the characteristic polynomial
PA(x).
Justify your answers.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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