If b is a basis for a subspace h, then each vector in h can be written in only one way as a linear combination of the vectors in b. the dimension of nul a is the number of variables in the equation ax = 0. the dimension of the column space of a is rank a. if b = {} is a basis for a subspace h of rn, then the correspondence makes h look and act the same as rp. if h is a p-dimensional subspace of rn, then a linearly independent set of p vectors in h is a basis for h.

Step-by-step explanation:

STEP 1 It's true

If H is a p-dimensional subspace of Rn, then B must be a set of p elements {v1, v2, . . . , vp}. To represent any vector b in H is to find the coefficients cj in the linear combination

b = c1v1 + c2v2 + . . . + cpvp

of vectors in B whose sum is b. This is equivalent to solving a system of equations with p equations and p unknowns. Because the vectors are linearly independent by the definition of a basis, this means there can only be one solution.

STEP 2 It's true

The dimension of Nul A is the number of free variables in the equation Ax = 0.

STEP 3 It's true

The rank of a matrix A is equal to the dimension of Col(A).

STEP 4 It's true

Because H is a p-dimensional subspace of Rn, any linearly independent set of exactly p elements in H is automatically a basis for H, hence also spans H

STEP 5 It's true

This correspondence is a one-to-one correspondence between H and Rp that preserves linear combinations, this is also known as an isomorphism. Because H is isomorphic to Rp, H looks and acts the same as Rp even though the vectors in H themselves may have more than p entries.

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕