To show that the map π1 : FN → F given by π1((an)) = a1 is linear, we need to show that it satisfies two properties: additivity and homogeneity.

1. Additivity: For any two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) in FN, we have:

π1(x + y) = π1((x1 + y1, x2 + y2, ..., xn + yn)) = x1 + y1 = π1(x) + π1(y)

2. Homogeneity: For any scalar c in F and any vector x = (x1, x2, ..., xn) in FN, we have:

π1(cx) = π1((cx1, cx2, ..., cxn)) = cx1 = cπ1(x)

Since π1 satisfies both additivity and homogeneity, we can conclude that π1 is a linear map.

Question

To show that the map π1 : FN → F given by π1((an)) = a1 is linear, we need to show that it satisfies two properties: additivity and homogeneity.

1. Additivity: For any two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) in FN, we have:

π1(x + y) = π1((x1 + y1, x2 + y2, ..., xn + yn)) = x1 + y1 = π1(x) + π1(y)

2. Homogeneity: For any scalar c in F and any vector x = (x1, x2, ..., xn) in FN, we have:

π1(cx) = π1((cx1, cx2, ..., cxn)) = cx1 = cπ1(x)

Since π1 satisfies both additivity and homogeneity, we can conclude that π1 is a linear map.

Knowee AI · Accepted Answer

To show that the map π1 : FN → F given by π1((an)) = a1 is linear, we need to show that it satisfies two properties: additivity and homogeneity.

1. Additivity: For any two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) in FN, we have:

π1(x + y) = π1((x1 + y1, x2 + y2, ..., xn + yn)) = x1 + y1 = π1(x) + π1(y)

2. Homogeneity: For any scalar c in F and any vector x = (x1, x2, ..., xn) in FN, we have:

π1(cx) = π1((cx1, cx2, ..., cxn)) = cx1 = cπ1(x)

Since π1 satisfies both additivity and homogeneity, we can conclude that π1 is a linear map.

1. Consider the map π1 ∶ FN Ð→ F given byπ1((an)) = a1.(a) Show that π1 is linear