In order for a transformation to be classified as linear, it must adhere to two key principles: vector addition and scalar multiplication. To establish that a transformation is indeed linear, one must verify two conditions. First, you should confirm that when you apply the transformation to the added value of two vectors, it results in the sum of the transformations of these individual vectors - a property known as additivity. Second, it's essential to ascertain that when a vector is multiplied by a scalar prior to the transformation, it is equivalent to the multiplication of the transformed vector by the same scalar - a condition referred to as homogeneity.
| Topic | Problem | Solution |
|---|---|---|
| None | Determine a linear transformation function that w… | Convert 105 degrees to radians: \(\theta_{rad} = 1.8326\) |
| None | (1 point) Find the matrix $A$ of the linear trans… | We are given a linear transformation $T(f(t))=8 f^{\prime}(t)+3 f(t)$ from $P_{2}$ to $P_{2}$ and w… |
| None | 3 Consider the map \( F: \mathbb{R}^{3} \rightarr… | Prove linearity: Let \( u=(x_1, y_1, z_1) \) and \( v=(x_2, y_2, z_2) \) in \( \mathbb{R}^{3} \) an… |