Showing certain elements in tensor product is linearly independent

Question

Let $A$ be a $k$-algebra. Let $l/k$ be a finite field ext. of $k$, and $b_1, ..., b_d$ be a $k$ basis of $l$. How can I show that $1 \otimes b_1, ..., 1 \otimes b_d \in A \otimes_k l$ is linearly independent over $A$? Thanks!

Answer 1

Pick any $k$-linear function $\epsilon:A\to k$ such that $\epsilon(1)=1$.

Now suppose there is a linear combination of your elements with coefficients in $k$ which is zero, and apply to it the linear map $\epsilon\otimes\mathrm{id}_l:A\otimes_kl\to k\otimes_kl=l$.

Answer 2

The map $L\longrightarrow A\otimes_K L$ in an injective ring homomorphism : hence if $\sum_{i=1}^d \lambda_i (1\otimes b_i)=\sum_{i=1}^d \lambda_i (1\otimes b_i)=1\otimes\sum_{i=1}^d ( \lambda_i b_i)=0$, then $\,\sum_{i=1}^d ( \lambda_i b_i)=0$, hence $\,\lambda_i=0$ for all $i=1,\dots,d$.