Sure, here is the R code for each part of your question:

(a) Compute ∥v1∥,∥v2∥, and v1 · v2.

```R
# Define the vectors
v1

Question

Sure, here is the R code for each part of your question:

(a) Compute ∥v1∥,∥v2∥, and v1 · v2.

```R
# Define the vectors
v1 <- c(1, 1, 1)
v2 <- c(1, 1, -1)

# Compute the norms
norm_v1 <- sqrt(sum(v1^2))
norm_v2 <- sqrt(sum(v2^2))

# Compute the dot product
dot_product <- sum(v1 * v2)
```

(b) Obtain vectors w1,w2, such that w1, w2 are orthogonal unit vectors, and span(v1, v2) = span(w1, w2).

```R
# Define w1 as v1 normalized
w1 <- v1 / norm_v1

# Define w2 as the orthogonal projection of v2 onto w1, normalized
w2 <- (v2 - sum(v2 * w1) * w1) / sqrt(sum((v2 - sum(v2 * w1) * w1)^2))
```

(c) Let w3 = (1/√2, −1/√2, 0)T . Check that w1 , w2 , w3 are mutually orthogonal unit vectors (i.e. they form an orthonormal basis of R3).

```R
# Define w3
w3 <- c(1/sqrt(2), -1/sqrt(2), 0)

# Check orthogonality
orthogonal <- all(abs(c(sum(w1 * w2), sum(w1 * w3), sum(w2 * w3))) < 1e-10)

# Check unit vectors
unit_vectors <- all(abs(c(sqrt(sum(w1^2)), sqrt(sum(w2^2)), sqrt(sum(w3^2))) - 1) < 1e-10)
```

(d) Define a 3×3 matrix A = [w1,w2,w3]. With the help of A, find linear coefficients x1, x2, x3, such that x1w1 + x2w2 + x3w3 = (3, 4, 5)⊤.

```R
# Define the matrix A
A <- cbind(w1, w2, w3)

# Define the vector (3, 4, 5)
v <- c(3, 4, 5)

# Solve the linear system
x <- solve(A, v)
```

The vector `x` contains the coefficients x1, x2, x3.

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