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sesenic [268]
4 years ago
11

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 = {vi,...,Vp} 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.
Mathematics
1 answer:
mixas84 [53]4 years ago
4 0

Answer:

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.

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1. -2/3        3. -5/4         5. 3/2          FOR THE REST OF THE PARTS LOOK THROUGH THE EXPLANATIONS

Step-by-step explanation:

Before I start the workings, I'll give a little explanation on this topic.

So, the equation of a straight line is

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where x and y are any coordinates of any point on the line.

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where rise is the how much a graph goes upwards or downwards, if its upwards use positive integer and if its downwards use negative integer.

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<u>MOVING TO ANSWERING NOW</u>

1. Dotted points on the line are (1,0) and (4,-2)

Formula of slope,

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I'll show both for this question

m = (y2-y1)/(x2-x1) = (-2-0)/(4-1) = -2/3

OR

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2. TRY DOING THIS ONE YOURSELF

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5.  (-8,-14) and (-10,-17)

m =  (y2-y1)/(x2-x1) = (-17-(-14))/(-10-(-8)) = -3/-2 = 3/2

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