find the components of the vector t(0,-3,4) in basis B

To find the components of a vector \(\mathbf{t} = (0, -3, 4)\) in the basis \(B = \{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) where:

\[
\mathbf{u} = (1, -1, 2), \quad \mathbf{v} = (1, 1, 1), \quad \mathbf{w} = (-1, 0, 1),
\]

we need to express \(\mathbf{t}\) as a linear combination of the basis vectors:

\[
\mathbf{t} = a \mathbf{u} + b \mathbf{v} + c \mathbf{w},
\]

where \(a, b,\) and \(c\) are the components of \(\mathbf{t}\) in the basis \(B\).

Substituting the values of \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), we have:

\[
(0, -3, 4) = a(1, -1, 2) + b(1, 1, 1) + c(-1, 0, 1).
\]

This equation gives us a system of linear equations:

\[
\begin{cases}
a + b - c = 0 \\
-a + b = -3 \\
2a + b + c = 4
\end{cases}
\]

### Step 1: Solve the System of Equations

Let's solve this system step-by-step.

1. **Equation 1**: \(a + b - c = 0\)
2. **Equation 2**: \(-a + b = -3\)
3. **Equation 3**: \(2a + b + c = 4\)

From **Equation 2**:

\[
b = a - 3.
\]

Substitute \(b = a - 3\) into **Equations 1** and **3**:

\[
\begin{cases}
a + (a - 3) - c = 0 \\
2a + (a - 3) + c = 4
\end{cases}
\]

Simplify the equations:

\[
\begin{cases}
2a - 3 - c = 0 \\
3a - 3 + c = 4
\end{cases}
\]

\[
\begin{cases}
2a - c = 3 \\
3a + c = 7
\end{cases}
\]

Add the two equations to eliminate \(c\):

\[
2a - c + 3a + c = 3 + 7 \implies 5a = 10 \implies a = 2.
\]

Now substitute \(a = 2\) back into \(b = a - 3\):

\[
b = 2 - 3 = -1.
\]

Finally, substitute \(a = 2\) and \(b = -1\) into \(2a - c = 3\):

\[
2(2) - c = 3 \implies 4 - c = 3 \implies c = 1.
\]

### Step 2: Write the Components in Basis \(B\)

The components of \(\mathbf{t} = (0, -3, 4)\) in the basis \(B = \{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) are:

\[
(a, b, c) = (2, -1, 1).
\]

Thus, the vector \(\mathbf{t} = (0, -3, 4)\) can be expressed in the basis \(B\) as:

\[
\mathbf{t} = 2\mathbf{u} - \mathbf{v} + \mathbf{w}.
\]

Would you like further clarification or have any other questions?

### Related Questions:

1. How do you find the components of a vector in a non-standard basis?
2. What are some examples of changing bases in higher dimensions?
3. How can we determine if two different sets of vectors form a basis for the same vector space?
4. What is the geometric interpretation of changing the basis of a vector space?
5. How do you compute the transformation matrix between two bases?

### Tip:

To verify the correctness of your solution, you can always substitute the calculated components back into the linear combination and check if it equals the original vector.

Learn how to find the components of vector t = (0, -3, 4) in basis B using linear algebra methods. This problem involves expressing the vector t as a linear combination of basis vectors u, v, and w. Detailed step-by-step solution provided.

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