All vectors and subspaces are in r^n.

check the true statements below:

a. for each y and each subspace w, the vector y - proj_w (y) is orthogonal to w.

b. if z is orthogonal to u_1 and u_2 and if w = span{u_1, u_2}. then z must be in w.

c. the orthogonal projection y of y onto a subspace w can sometimes depend on the orthogonal basis for w used to compute y.

d. if y is in a subspace w, then the orthogonal projection of y onto w is y itself.

e. if the columns of an n times p matrix u are orthonormal, then uu^t y is the orthogonal projection of y onto the column space of u

answer: do we write it

step-by-step explanation:

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)

168

step-by-step explanation:

first off, we know that he drove 3.5 hours at a rate of 48 mph.

knowing that, we want to see how many miles he drove all-together.

in this case, we can just multiply 48 by 3.5.

48 * 3.5 = 168

for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis: