How many Hecke operators span the Hecke algebra?

I sort of understand what "span" mean.....My understanding is that if a augumented matrix has a solution. Ax=b's b part spans Matrix A But what I don't get is when the following question is asked. Determine if following Matrix spans R^4...what...

Describe all linear operators T∈Hom(R^2, R^2) such that T is diagonalizable and T^3-2T^2+T = 0.

Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination...

Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination...

Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination...

Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination...

Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination...

why is the span S of any set of vectors in a vector space V a subspace of V?

Let V ve a vector space over a field F A) let x_1.....x_n and y_1......y_n be in V Show that Span(x_1...x_n, y_1....y_x)=Span(x_1..x_n) + Span(y_1....y_n) b) let x_1, x_2, x_3, x_4 be four linearly independent vectors in V. Show that Span(x_1,x_2,x_...