Siyabonga Sanele Mdluli
May 28
Examples of using base vectors to find matrix of transformation?
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ChatGPT
May 28
1. Suppose we have a transformation T: R^2 -> R^2 defined by T(x, y) = (2x + y, x - y). To find the matrix of this transformation with respect to the standard basis vectors e1 = (1, 0) and e2 = (0, 1), we can apply T to each basis vector and express the result as a linear combination of the basis vectors:
T(e1) = T(1, 0) = (2*1 + 0, 1 - 0) = (2, 1) = 2e1 + e2
T(e2) = T(0, 1) = (2*0 + 1, 0 - 1) = (1, -1) = e1 - e2
Therefore, the matrix of the transformation T with respect to the standard basis vectors is:
[2 1]
[1 -1]
2. Consider a transformation T: R^3 -> R^3 defined by T(x, y, z) = (x + y, y - z, x + z). To find the matrix of this transformation with respect to the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1), we can apply T to each basis vector and express the result as a linear combination of the basis vectors:
T(e1) = T(1, 0, 0) = (1 + 0, 0 - 0, 1 + 0) = (1, 0, 1) = e1 + e3
T(e2) = T(0, 1, 0) = (0 + 1, 1 - 0, 0 + 0) = (1, 1, 0) = e1 + e2
T(e3) = T(0, 0, 1) = (0 + 0, 0 - 1, 0 + 1) = (0, -1, 1) = -e2 + e3
Therefore, the matrix of the transformation T with respect to the standard basis vectors is:
[1 1 0]
[0 1 -1]
[1 0 1]
T(e1) = T(1, 0) = (2*1 + 0, 1 - 0) = (2, 1) = 2e1 + e2
T(e2) = T(0, 1) = (2*0 + 1, 0 - 1) = (1, -1) = e1 - e2
Therefore, the matrix of the transformation T with respect to the standard basis vectors is:
[2 1]
[1 -1]
2. Consider a transformation T: R^3 -> R^3 defined by T(x, y, z) = (x + y, y - z, x + z). To find the matrix of this transformation with respect to the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1), we can apply T to each basis vector and express the result as a linear combination of the basis vectors:
T(e1) = T(1, 0, 0) = (1 + 0, 0 - 0, 1 + 0) = (1, 0, 1) = e1 + e3
T(e2) = T(0, 1, 0) = (0 + 1, 1 - 0, 0 + 0) = (1, 1, 0) = e1 + e2
T(e3) = T(0, 0, 1) = (0 + 0, 0 - 1, 0 + 1) = (0, -1, 1) = -e2 + e3
Therefore, the matrix of the transformation T with respect to the standard basis vectors is:
[1 1 0]
[0 1 -1]
[1 0 1]
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Google Gemini AI
May 28
Example 1: Translation Transformation
- Original Coordinates: (x, y)
- Translation Vector: (tx, ty)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
T = [1 0 0]
[0 1 0]
[tx ty 1]
```
Explanation:
The translation transformation matrix T moves a point (x, y) by (tx, ty) units. The first two rows of T represent the unit vectors i and j, respectively, which do not change under translation. The third row contains the translation vector (tx, ty) that is added to the original coordinates.
Example 2: Scaling Transformation
- Original Coordinates: (x, y)
- Scaling Factors: (sx, sy)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
S = [sx 0 0]
[0 sy 0]
[0 0 1]
```
Explanation:
The scaling transformation matrix S scales a point (x, y) by sx and sy in the x and y directions, respectively. The first two rows of S contain the scaling factors sx and sy multiplied by the unit vectors i and j. The third row is the identity vector (0, 0, 1) to maintain the homogeneity of the coordinates.
Example 3: Rotation Transformation
- Original Coordinates: (x, y)
- Rotation Angle: θ
- Base Vectors: i = (cos θ, sin θ), j = (-sin θ, cos θ)
- Transformation Matrix:
```
R = [cos θ sin θ 0]
[-sin θ cos θ 0]
[0 0 1]
```
Explanation:
The rotation transformation matrix R rotates a point (x, y) by θ radians around the origin. The first two rows of R contain the new base vectors i and j that are rotated by θ. The third row is the identity vector to maintain the homogeneity of the coordinates.
Example 4: Shear Transformation
- Original Coordinates: (x, y)
- Shear Factors: (shx, shy)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
Sh = [1 shx 0]
[shy 1 0]
[0 0 1]
```
Explanation:
The shear transformation matrix Sh shears a point (x, y) by shx and shy in the x and y directions, respectively. The first two rows of Sh contain the shear factors shx and shy added to the unit vectors i and j. The third row is the identity vector to maintain the homogeneity of the coordinates.
- Original Coordinates: (x, y)
- Translation Vector: (tx, ty)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
T = [1 0 0]
[0 1 0]
[tx ty 1]
```
Explanation:
The translation transformation matrix T moves a point (x, y) by (tx, ty) units. The first two rows of T represent the unit vectors i and j, respectively, which do not change under translation. The third row contains the translation vector (tx, ty) that is added to the original coordinates.
Example 2: Scaling Transformation
- Original Coordinates: (x, y)
- Scaling Factors: (sx, sy)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
S = [sx 0 0]
[0 sy 0]
[0 0 1]
```
Explanation:
The scaling transformation matrix S scales a point (x, y) by sx and sy in the x and y directions, respectively. The first two rows of S contain the scaling factors sx and sy multiplied by the unit vectors i and j. The third row is the identity vector (0, 0, 1) to maintain the homogeneity of the coordinates.
Example 3: Rotation Transformation
- Original Coordinates: (x, y)
- Rotation Angle: θ
- Base Vectors: i = (cos θ, sin θ), j = (-sin θ, cos θ)
- Transformation Matrix:
```
R = [cos θ sin θ 0]
[-sin θ cos θ 0]
[0 0 1]
```
Explanation:
The rotation transformation matrix R rotates a point (x, y) by θ radians around the origin. The first two rows of R contain the new base vectors i and j that are rotated by θ. The third row is the identity vector to maintain the homogeneity of the coordinates.
Example 4: Shear Transformation
- Original Coordinates: (x, y)
- Shear Factors: (shx, shy)
- Base Vectors: i = (1, 0), j = (0, 1)
- Transformation Matrix:
```
Sh = [1 shx 0]
[shy 1 0]
[0 0 1]
```
Explanation:
The shear transformation matrix Sh shears a point (x, y) by shx and shy in the x and y directions, respectively. The first two rows of Sh contain the shear factors shx and shy added to the unit vectors i and j. The third row is the identity vector to maintain the homogeneity of the coordinates.
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